Geometry Marathon Autors: Mathlinks Forum Edited by Ercole Suppa1 July 14, 2012
1. Inradius of a triangle, with integer sides, is equal to 1. Find the sides of the triangle and prove that one of its angle is 90◦ . 2. Let O be the circumcenter of an acute triangle ABC and let k be the circle with center S that is tangent to O at A and tangent to side BC at D. Circle k meets AB and AC again at E and F respectively. The lines OS and ES meet k again at I and G. Lines BO and IG intersect at H. Prove that DF 2 . GH = AF 3. ABCD is parellelogram and a straight line cuts AB at AD 4 and AC at x · AC. Find x.
AB 3
and AD at
4. In 4ABC, ∠BAC = 120◦ . Let AD be the angle bisector of ∠BAC. Express AD in terms of AB and BC. 5. In a triangle ABC, AD is the feet of perpendicular to BC. The inradii of ADC, ADB and ABC are x, y, z. Find the relation between x, y, z. 6. Prove that the third pedal triangle is similar to the original triangle. 7. ABCDE is a regular pentagon and P is a point on the minor arc AB. Prove that P A + P B + P D = P C + P E. 8. Two congruent equilateral triangles, one with red sides and one with blue sides overlap so that their sides intersect at six points, forming a hexagon. If r1 , r2 , r3 , b1 , b2 , b3 are the red and blue sides of the hexagon respectively, prove that (a) r12 + r22 + r32 = b21 + b22 + b23 (b) r1 + r2 + r3 = b1 + b2 + b3 9. if in a quadrilateral ABCD, AB + CD = BC + AD. Prove that the angle bisectors are concurrent at a point which is equidistant from the sides of the sides of the quadrilateral. 1 Email:
[email protected],
Web: http://www.esuppa.it/
1
10. In a triangle with sides a, b, c, let r and R be the inradius and circumradius respectively. Prove that for all such non-degenerate triangles, 2rR =
abc a+b+c
11. Prove that the area of any non degenerate convex quadrilateral in the cartesian plane which has an incircle is given by ∆ = rs where r is the inradius and s is the semiperimeter of the polygon. 12. Let ABC be a equilateral triangle with side a. M is a point such that M S = d, where S is the circumcenter of ABC. Prove that the area of the triangle whose sides are M A, M B, M C is √ 2 3|a − 3d2 | 12 13. Prove that in a triangle, SI12 = R2 + 2Rra 14. Find the locus of P in a triangle if P A2 = P B 2 + P C 2 . 15. 16. In an acute triangle ABC, let the orthocenter be H and let its projection on the median from A be X. Prove that BHXC is cyclic. 17. If ABC is a right triangle with A = 90◦ , if the incircle meets BC at X, prove that [ABC] = BX · XC. 18. n regular polygons in a plane are such that they have a common vertex O and they fill the space around O completely. The n regular polygons have a1 , a2 , · · · , an sides not necessarily in that order. Prove that n X 1 n−2 = a 2 i=1 i
19. Let the equation of a circle be x2 + y 2 = 100. Find the number of points (a, b) that lie on the circle such that a and b are both integers. 20. S is the circumcentre of the 4ABC. 4DEF is the orthic triangle of 4ABC. Prove that SA is perpendicular to EF , SB is the perpendicular to DF and SC is the perpendicular to DE. 21. ABCD is a parallelogram and P is a point inside it such that ∠AP B + ∠CP D = 180◦ . Prove that AP · CP + BP · DP = AB · BC 2
22. ABC is a non degenerate equilateral triangle and P is the point diametrically opposite to A in the circumcircle. Prove that P A × P B × P C = 2R3 where R is the circumradius. 23. In a triangle, let R denote the circumradius, r denote the inradius and A denote the area. Prove that: √ 9r2 ≤ A 3 ≤ r(4R + r) with equality if, and only if, the triangle is equilateral. 23. If in a triangle, O, H, I have their usual meanings, prove that 2 · OI ≥ IH 24. In acute angled triangle ABC, the circle with diameter AB intersects the altitude CC 0 and its extensions at M and N and the circle with diameter AC intersects the altitude BB 0 and its extensions at P and Q. Prove that M , N , P , Q are concyclic. 25. Given circles C1 and C2 which intersect at points X and Y , let `1 be a line through the centre of C1 which intersects C2 at points P , Q. Let `2 be a line through the centre of C2 which intersects C1 at points R, S. Show that if P , Q, R, S lie on a circle then the centre of this circle lies on XY . 26. From a point P outside a circle, tangents are drawn to the circle, and the points of tangency are B, D. A secant through P intersects the circle at A, C. Let X, Y , Z be the feet of the altitudes from D to BC, A, AB respectively. Show that XY = Y Z. 27. 4ABC is acute and ha , hb , hc denote its altitudes. R, r, r0 denote the radii of its circumcircle, incircle and incircle of its orthic triangle (whose vertices are the feet of its altitudes). Prove the relation: ha + hb + hc = 2R + 4r + r0 +
r2 R
28. In a triangle 4ABC, points D, E, F are marked on sides BC, CA, AC, respectively, such that BD CE AF = = =2 DC EA FB Show that (a) The triangle formed by the lines AD, BE, CF has an area 1/7 that of 4ABC. (b) (Generalisation) If the common ratio is k (greater than 1) then the 2 triangle formed by the lines AD, BE, CF has an area k(k−1) 2 +k+1 that of 4ABC. 3
29. Let AD , the altitude of 4ABC meet the circum-circle at D0 . Prove that the Simson’s line of D0 is parallel to the tangent drawn from A. 30. Point P is inside 4ABC. Determine points D on side AB and E on side AC such that BD = CE and P D + P E is minimum. 31. Prove this result analogous to the Euler Line. In triangle 4ABC, let G, I, N be the centroid, incentre, and Nagel point, respectively. Show that, (a) I, G, N lie on a line in that order, and that N G = 2 · IG. (b) If P, Q, R are the midpoints of BC, CA, AB respectively, then the incentre of 4P QR is the midpoint of IN . 32. The cyclic quadrialteral ABCD satisfies AD + BC = AB. Prove that the internal bisectors of ∠ADC and ∠BCD intersect on AB. 33. Let ` be a line through the orthocentre H of a triangle 4ABC. Prove that the reflections of ` across AB, BC, CA pass through a common point lying on the circumcircle of 4ABC. 34. If circle O with radius r1 intersect the sides of triangle ABC in six points. Prove that r1 ≥ r, where r is the inradius. 35. Construct a right angled triangle given its hypotenuse and the fact that the median falling on hypotenuse is the geometric mean of the legs of the triangle. √ 36. Find the angles of the triangle which satisfies R(b + c) = a bc where a, b, c, R are the sides and the circumradius of the triangle. 37. (MOP 1998) Let ABCDEF be a cyclic hexagon with AB = CD = EF . Prove that the intersections of AC with BD, of CE with DF , and of EA with F B form a triangle similar to 4BDF . 38. 4ABC is right-angled and assume that the perpendicular bisectors of BC, CA, AB cut its incircle (I) at three chords. Show that the lenghts of these chords form a right-angled triangle. 38. We have a trapezoid ABCD with the bases AD and BC. AD = 4, BC = 2, AB = 2. Find possible values of ∠ACD. 39. Find all convex polygons such that one angle is greater than the sum of the other angles. 40. If A1 A2 A3 · · · An is a regular n-gon and P is any point on its circumcircle, then prove that (i) P A21 + P A22 + P A23 + · · · + P A2n is constant; (ii) P A41 + P A42 + P A43 + · · · + P A4n is constant.
4
41. In a triangle ABC the incircle γ touches the sides BC, CA,AD at D, E, F respectively. Let P be any point within γ and let the segments AP , BP , CP meet γ at X, Y , Z respectively. Prove that DX, EY , F Z are concurrent. 42. ABCD is a convex quadrilateral which has incircle (I, r) and circumcircle (O, R), show that: 2R2 ≥ IA · IC + IB · ID ≥ 4r2 43. Let P be any point in 4ABC. Let AP , BP , CP meet the circumcircle of 4ABC again at A1 , B1 , C1 respectively. A2 , B2 , C2 are the reflections of A1 , B1 , C1 about the sides BC, AC, AB respectively. Prove that the circumcircle of 4A2 B2 C2 passes through a fixed point independent of P . 44. A point P inside a circle is such that there are three chords of the same length passing through P . Prove that P is the center of the circle. 45. ∆ABC is right-angled with ∠BAC = 90◦ . H is the orthogonal projection of A on BC. Let r1 and r2 be the inradii of the triangles 4ABH and 4ACH. Prove q AH = r1 + r2 +
r12 + r22
46. Let ABC be a right angle triangle with ∠BAC = 90◦ . Let D be a point on BC such that the inradius of 4BAD is the same as that of 4CAD. Prove that AD2 is the area of 4ABC. 47. τ is an arbitrary tangent to the circumcircle of 4ABC and X, Y , Z are the orthogonal projections of A, B, C on τ . Prove that with appropiate choice of signs we have: √ √ √ ±BC AX ± CA BY ± AB CZ = 0 48. Let ABCD be a convex quadrilateral such that AB + BC = CD + DA. Let I, J be the incentres of 4BCD and 4DAB respectively. Prove that AC, BD, IJ are concurrent. 49. 4ABC is equilateral with side lenght L. P is a variable point on its incircle and A0 , B 0 , C 0 are the orthogonal projections of P onto BC, CA, AB. Define ω1 , ω2 , ω3 as the circles tangent to the circumcircle of 4ABC at its minor arcs BC, CA, AB and tangent to BC, CA, AB at A0 , B 0 , C 0 respectively. δij stands for the lenght of the common external tangent of the circles ωi , ωj . Show that δ12 + δ23 + δ31 is constant and compute such value. 50. It is given a triangle 4ABC with AB 6= AC. Construct a tangent line τ to its incircle (I) which meets AC, AB at X, Y such that: AX AY + = 1. XC YB 5
51. In 4ABC, AB + AC = 3 · BC. Let the incentre be I and the incircle be tangent to AB, AC at D, E respectively. Let D0 , E 0 be the reflections of D, E about I. Prove that BCD0 E 0 is cyclic. 52. 4ABC has incircle (I, r) and circumcircle (O, R). Prove that, there exists a common tangent line to the circumcircles of 4OBC, 4OCA and 4OAB if and only if: R √ = 2+1 r 53. In a 4ABC,prove that a · AI 2 + b · BI 2 + c · CI 2 = abc 54. In cyclic quadrilateral ABCD, ∠ABC = 90◦ and AB = BC. If the area of ABCD is 50, find the length BD. 55. Given four points A, B, C, D in a straight line, find a point O in the same straight line such that OA : OB = OC : OD. 56. Let the incentre of 4ABC be I and the incircle be tangent to BC, AC at E, D. Let M , N be midpoints of AB, AC. Prove that BI, ED, M N are concurrent. 57. let O and H be circumcenter and orthocenter of ABC respectively. The perpendicular bisector of AH meets AB and AC at D and E respectively. Show that ∠AOD = ∠AOE. 58. Given a semicircle with diameter AB and center O and a line, which intersects the semicircle at C and D and line AB at M (M B < M A, M D < M C). Let K be the second point of intersection of the circumcircles of 4AOC and 4DOB. Prove that ∠M KO = 90◦ . 59. In the trapezoid ABCD, AB k CD and the diagonals intersect at O. P , Q are points on AD and BC respectively such that ∠AP B = ∠CP D and ∠AQB = ∠CQD. Show that OP = OQ. 60. In cyclic quadrilateral ABCD, ∠ACD = 2∠BAC and ∠ACB = 2∠DAC. Prove that BC + CD = AC. 61. 4ABC is right with hypotenuse BC. P lies on BC and the parallels through P to AC, AB meet the circumferences with diameters P C, P B again at U , V respectively. Ray AP cuts the circumcircle of 4ABC at D. Show that ∠U DV = 90◦ . 62. Let ABEF and ACGH be squares outside 4ABC. Let M be the midpoint of EG. Show that 4M BC is an isoceles right triangle. 63. The three squares ACC1 A00 , ABB10 A0 , BCDE are constructed externally on the sides of a triangle ABC. Let P be the center of BCDE. Prove that the lines A0 C, A00 B, P A are concurrent. 6
64. For triangle ABC, AB < AC, from point M in AC such that AB +AM = M C. The straight line perpendicular AC at M cut the bisection of BC in I. Call N is the midpoint of BC. Prove that is M N perpendicular to the AI. 65. Let ABC be a triangle with AB 6= AC. Point E is such that AE = BE and BE ⊥ BC. Point F is such that AF = CF and CF ⊥ BC. Let D be the point on line BC such that AD is tangent to the circumcircle of triangle ABC. Prove that D, E, F are collinear. 66. Points D, E, F are outside triangle ABC such that ∠DBC = ∠F BA, ∠DCB = ∠ECA, ∠EAC = ∠F AB. Prove that AD, BE, CF are concurrent. 67. In 4ABC, ∠C = 90◦ , and D is the perpendicular from C to AB. ω is the circumcircle of 4BCD. ω1 is a circle tangent to AC, AB, and ω. Let M be the point of tangency of ω1 with AB. Show that BM = BC. 68. Acute triangle 4ABC has orthocenter H and semiperimeter s. ra , rb , rc denote its exradii and %a , %b , %c denote the inradii of triangles 4HBC, 4HCA and 4HAB. Prove that: ra + rb + rc + %a + %b + %c = 2s 69. The lengths of the altitudes of a triangle are 12,15,20. Find the sides of the triangle and the area of the triangle? 70. Suppose, in an obtuse angled triangle, the orthic triangle is similar to the original triangle. What are the angles of the obtuse triangle? 71. In triangle ∆ABC with semiperimeter s, the incircle (I, r) touches side BC in X. If h represents the lenght of the altitude from vertex A to BC. Show that AX 2 = 2r.h + (s − a)2 72. Let E, F be on AB, AD of a cyclic quadrilateral ABCD such that AE = CD and AF = BC. Prove that AC bisects the line EF . 73. Suppose X and Y are two points on side BC of triangle ABC with the following property: BX = CY and ∠BAX = ∠CAY . Prove AB = AC. 74. ABC is a triangle in which I is its incenter. The incircle is drawn and 3 tangents are drawn to the incircle such that they are parellel to the sides of ABC. Now, three triangle are formed near the vertices and their incircles are drawn. Prove that the sum of the radii of the three incircles is equal to the radius of the the incircle of ABC. 75. With usual notation of I, prove that the Euler lines of 4IBC, 4ICA, 4IAB are concurrent. 7
76. Vertex A of 4ABC is fixed and B, C move on two fixed rays Ax, Ay such that AB + AC is constant. Prove that the loci of the circumcenter, centroid and orthocenter of 4ABC are three parallel lines. 77. 4ABC has circumcentre O and incentre I. The incentre touches BC, AC, AB at D, E, F and the midpoints of the altitudes from A, B, C are P , Q, R. Prove that DP , EQ, F R, OI are concurrent. 78. The incircle Γ of the equilateral triangle 4ABC is tangent to BC, CA, AB at M , N , L. A tangent line to Γ through its minor arc N L cut AB, AC at P , Q. Show that: 1 6 1 + = [M P B] [M QC] [ABC] 79. A and B are on a circle with center O such that AOB is a quarter of the circle. Square OEDC is inscribed in the quarter circle, with E on OB, D on the circle, and C on OA. Let F be on arc AD such that CDbisects∠F CB. Show that BC = 3 · CF . 80. Take a circle with a chord drawn in it, and consider any circle tangent to both the chord and the minor arc. Let the point of tangency for the small circle and the chord be X. Also, let the point of tangency for the small circle and the minor arc be Y . Prove that all lines XY are concurrent. 81. Two circles intersect each other at A and B. Line P T is a common tangent, where P and T are the points of tangency. Let S be the intersection of the two tangents to the circumcircle of 4AP T at P and T . Let H be the reflection of B over P T . Show that A, H, and S are collinear. 82. In convex hexagon ABCDEF , AD = BC + EF , BE = CD + AF and CF = AB + DE. Prove that AB CD EF = = . DE AF BC 83. The triangle ABC is scalene with AB > AC. M is the midpoint of BC and the angle bisector of ∠BAC hits the segment BC at D. N is the perpendicular foot from C to AD. Given that M N = 4 and DM = 2. Compute the value AM 2 − AD2 . 84. A, B, C, and D are four points on a line, in that order. Isoceles triangles AEB, BF C, and CGD are constructed on the same side of the line, with AE = EB = BF = F C = CG = GD. H and I are points so that BEHF and CF IG are rhombi. Finally, J is a point such that F HJI is a rhombus. Show that JA = JD. 85. A line through the circumcenter O of 4ABC meets sides AB and AC at M and N , respectively. Let R and S be the midpoints of CM and BN respectively. Show that ∠BAC = ∠ROS. 8
86. Let AB be a chord in a circle and P a point on the circle. Let Q be the foot of the perpendicular from P to AB, and R and S the feet of the perpendiculars from P to the tangents to the circle at A and B. Prove that P Q2 = P R · P S. 87. Given a circle ω with diameter AB, a line outside the circle d is perpendicular to AB closer to B than A. C ∈ ω and D = AC ∩ d. A tangent from D is drawn to Eonω such that B, E lie on same side of AC. F = BE ∩ d and G = F A ∩ ω and G0 = F C ∩ ω. Show that the reflection of G across AB is G0 . 88. 4ABC is acute and its angles α, β, γ are measured in radians. S and S0 represent the area of 4ABC and the area bounded/overlapped by the three circles with diameters BC, CA, AB respectively. Show that: S + 2S0 =
b2 π c2 π a2 π −α + −β + −γ 2 2 2 2 2 2
89. Let 4ABC be an isosceles triangle with AB = AC and ∠A = 30◦ . The triangle is inscribed in a circle with center O. The point D lies on the arch between A and C such that ∠DOC = 30◦ . Let G be the point on the arch between A and B such that AC = DG and AG < BG. The line DG intersects AC and AB in E and F respectively. (a) Prove that 4AF G is equilateral. (b) Find the ratio between the areas
4AGF 4ABC .
90. Construct a triangle ABC given the lengths of the altitude, median and inner angle bisector emerging from vertex A. 91. Let P be a point in 4ABC such that ∠P AC = ∠P BA + ∠P CA.
AB BC
=
AP PC .
Prove that ∠P BC +
92. Point D lies inside the equilateral 4ABC, such that DA2 = DB 2 + DC 2 . Show that ∠BDC = 150◦ . 93. (China MO 1998) Find the locus of all points D with respect to a given triangle 4ABC such that DA · DB · AB + DB · DC · BC + DC · DA · CA = AB · BC · CA. 94. Let P be a point in equilateral triangle ABC. If ∠BP C = α, ∠CP A = β, ∠AP B = γ, find the angles of the triangle with side lengths P A, P B, P C. 95. Of a ABCD, let P, Q, R, S be the midpoints of the sides AB, BC, CD, DA. Show that if 4AQR and 4CSP are equilateral, then ABCD is a rhombus. Also find its angles.
9
96. In ∆ABC, the incircle touches BC at the point X. A0 is the midpoint of BC. I is the incentre of ∆ABC. Prove that A0 I bisects AX. 97. In convex quadrilateral ABCD, ∠BAC = 80◦ , ∠BCA = 60◦ , ∠DAC = 70◦ , ∠DCA = 40◦ . Find ∠DBC. 98. It is given a 4ABC and let X be an arbitrary point inside the triangle. If XD⊥AB, XE⊥BC, XF ⊥AC, where D ∈ AB, E ∈ BC, F ∈ AC, then prove that: AX + BX + CX ≥ 2(XD + XE + XF ) 99. Let A1 , A2 , A3 and A4 be four circles such that the circles A1 and A3 are tangential at a point P , and the circles A2 and A4 are also tangential at the same point P . Suppose that the circles A1 and A2 meet at a point T1 , the circles A2 and A3 meet at a point T2 , the circles A3 and A4 meet at a point T3 , and the circles A4 and A1 meet at a point T4 , such that all these four points T1 , T2 , T3 , T4 are distinct from P . Prove that
T1 T2 T1 T4
2 T2 T3 P T2 · = T3 T4 P T4
100. ABCD is a convex quadrilateral such that ∠ADB + ∠ACB = 180◦ . It’s diagonals AC and BD intersect at M . Show that AB 2 = AM · AC + BM · BD 101. Let AH, BM be the altitude and median of triangle ABC from A and B. If AH = BM , find ∠M BC. 102. P , Q, R are random points in the interior of BC, CA, and AB respectively of a non-degenerate triangle ABC such that the circumcircles of BP R and CQP are orthogonal and intersect in M other than P . Prove that P R · M Q, P Q · M R, QR · M P can be the sides of a right angled triangle. 103. 4ABC is scalene and D is a point on the arc BC of its circumcircle which doesn’t contain A. Perpendicular bisectors of AC, AB cut AD at Q, R. If P ≡ BR ∩ CQ, then show that AD = P B + P C. 104. It is given 4ABC and M is the midpoint of the segment AB. Let ` pass through M and ` ∩ AC = K and ` ∩ BC = L, such that CK = CL. Let CD⊥AB, D ∈ AB and O is the center of the circle, circumscribed around 4CKL. Prove that OM = OD. 105. Prove that: The locus of points P in the plane of an acute triangle 4ABC which satisfy that the lenght of segments P A, P B, P C can form a right triangle is the union of three circumferences, whose centers are the reflections of A, B, midpoints of BC, √ C across the √ √ CA, AB and whose radii are given by b2 + c2 − a2 , a2 + c2 − b2 , a2 + b2 − c2 . 10
106. Let D, E be points on the rays BA, CA respectively such that BA · BD + CA · CE = BC 2 . Prove that ∠CDA = ∠BEC. 107. In triangle ABC, M , N , P are points on sides BC, CA, AB respectively such that perimeter of the triangle M N P is minimal. Prove that triangle M N P is the orthic triangle of ABC (the triangle formed by the foot of the perpendiculars on the sides as vertices). 108. Prove that there exists an inversion mapping two non-intersecting circles into concentric circles. 109. Let α, β, γ be three circles concurring at M . AM , BM , CM are the common chords of α, β; β, γ; and γ, α respectively. AM , BM , CM intersect γ, α, β at P , Q, R respectively. Prove that AQ · BR · CP = AR · BP · CQ 110. In triangle 4ABC, lines `b and `c are perpendicular to BC through vertices B, C respectively. P is a variable point on line BC and the perpendicular lines dropped from P to AB, AC cut `b , `c at U , V respectively. Show that U V always passes through the orthocenter of 4ABC. 111. Let I be the incenter of triangle ABC and M is the midpoint of BC. The excircle opposite A touches the side BC at D. Prove that AD k IM . 112. An incircle of ABC triangle tangents BC, CA and AB sides at A1 , B1 and C1 points, respectively. Let O and I be circumcenter and incenter and OI ∩ BC = D. A line through A1 point and perpendicular to B1 C1 cut AD at E. Prove that M point lies on B1 C1 line. (M is midpoint of EA1 ). 113. Parallels are drawn to the sides of the triangle ABC such that the lines touch the in-circle of ABC. The lengths of the tangents within ABC are x, y, z respectively opposite to sides a, b, c respectively. Prove the relation: x y z + + =1 a b c 114. In an acute angled triangle ABC, the points D, E, F are on sides BC, CA, AB respectively, such that ∠AF E = ∠BF D, ∠F DB = ∠EDC, ∠DEC = ∠F EA. Prove that DEF is the orthic triangle of ABC. 115. Let ω be circle and tangents AB, AC sides and circumcircle/internally and at D point. Prove that circumcenter of 4ABC lies on bisector of ∠BDC. 116. Construct a triangle with ruler-compass operations, given its inradius, circumradius and any altitude. 117. Let AD, BM , CH be the angle bisector, median, altitude from A, B, C of 4ABC. If AD = BM = CH, prove that 4ABC is equilateral. 11
118. Consider a triangle ABC with BC = a, CA = b, AB = c and area equal to 4. Let x, y, z√the distances√ from the √ orthocenter to the vertices A, B, C. √ Prove that if a x + b y + c z = 4 a + b + c, then ABC is equilateral. 119. Suppose that ∠A is the smallest of the three angles of triangle ABC. Let D be a point on the arc BC of the circumcircle of 4ABC not containing A. Let the perpendicular bisectors of AB, AC intersect AD at M and N respectively. Let BM and CN meet at T . Prove that BT + CT ≤ 2R where R is the circumradius of triangle ABC. 120. Points E, F are taken on the side AB of triangle ABC such that the lengths of CE and CF are both equal to the semiperimeter of the triangle ABC. Prove that the circumcircle of CEF is tangent to the excircle of triangle ABC opposite C. 121. Two fixed circles ω1 , ω2 intersect at A, B. A line ` through A cuts ω1 , ω2 again at U , V . Show that the perpendicular bisector of U V goes through a fixed point as line ` spins around A. 122. Let 4ABC be an isosceles triangle with AB = AC. Let X and Y be points on sides BC and CA such that XY k AB. Let D be the circumcenter of 4CXY and E be the midpoint of BY . Prove that ∠AED = 90◦ . 123. Tetrahedron ABCD is √ featured on ball (centre S, r = 1) and SA ≥ SB ≥ SD. Prove that SA > 5. 124. Let ABCD be a cyclic quadrilateral. The lines AB and CD intersect at the point E, and the diagonals AC and BD at the point F . The circumcircle of the triangles AF D and BF C intersect again at H. Prove that EHF = 90◦ . 125. ABCD is a cyclic and circumscribed quadrilateral whose incircle touches the sides AB, BC, CD, DA at E, F , G, H. Prove that EG ⊥ F H. 126. Let τ be an arbitrary tangent line to the circumcircle (O, R) of 4ABC. δ(P ) stands for the distance from point P to τ . If I, Ia , Ib , Ic denote the incenter and the three excenters of 4ABC, prove with appropiate choice of signs that: ±δ(I) ± δ(Ia ) ± δ(Ib ) ± δ(Ic ) = 4R 127. Let ABC be a fixed triangle and β, γ are fixed angles. Let α be a variable angle. Let E, F be points outside 4ABC such that ∠F BA = β, ∠F AB = α, ∠ECA = γ, ∠EAC = α. Prove that the intersection of BE, CF lies on a fixed line independent of α. 128. Incircle (I) of 4ABC touches BC, CA, AB at D, E, F and BI, CI cut CA, AB at M , N . Line M N intersects (I) at two points, let P be one of these points. Show that the lengths of segments P D, P E, P F form a right triangle. 12
129. Given a triangle ABC with orthocentre H, circumcentre O, incentre I and D is the tangency point of incircle with BC. Prove that if OI and BC are parallel, then AO and HD are parallel as well. AB AD 130. Let ABCD be a cyclic quadrilateral such that BC = DC . The circle passing through A, B and tangent to AD intersects CB at E. The circle passing through A, D and tangent to AB intersects CD at F . Prove that BEF D is cyclic.
131. Given two points A, B and a circle (O) not containing A, B. Consider the radical axis of an arbitrary circle passing through A, B and (O). Prove that all such radical axes passes through a fixed point P and construct it. 132. Given a sphere of radius one that tangents the six edges of an arbitrary tetrahedron. Find the maximum possible volume of the tetrahedron. 133. Let ABC be a triangle for which exists D ∈ BC so that AD ⊥ BC. Denote r1 , r2 the lengths of inradius for the triangles ABD, ADC respectively. Prove that ar1 + (s − a)(s − c) = ar2 + (s − a)(s − b) = sr 134. Let M be the midpoint of BC of triangle ABC. Suppose D is a point on AM . Prove that ∠DBC = ∠DAB if and only if ∠DCB = ∠DAC. 135. P and R are two given points on a circle Ω. Let O be an arbitrary point on the perpendicular bisector of P R. A circle with centre O intersects OP and OR at the points M , N respectively. The tangents to this circle at M and N meet ω at points Q and S respectively such that P , Q, R, S lie on Ω in this order. P Q and RS intersect at K. Show that the line joining the midpoints of P Q and RS is perpendicular to OK. 136. In cyclic quadrilateral ABCD, AB = 8, BC = 6, CD = 5, DA = 12. Let AB intersect DC at E. Find the length EB. 137. In triangle ABC, let Γ be a circle passing through B and C and intersecting AB and AC at M , N respectively. Prove that the locus of the midpoint of M N is the A-symmedian of the triangle. 138. Let ABC be a triangle E is the excenter of 4ABC opposite A. If AC + CB = AB + BE, find ∠ABC. 139. In a given line segment AB, choose an arbitrary point C in the interior. The point D, E, F are the midpoints of the segments AC, CB and AB respectively, and consider the point X in the interior of the line segment CF such that CX F X = 2. Prove that BX AX = =2 DX XE 13
140. Diagonals of a convex quadrilateral with an area of Q divide it into four triangles with appropriate areas P1 , P2 , P3 , P4 . Prove that P1 · P2 · P3 · P4 =
(P1 + P2 )2 · (P2 + P3 )2 · (P3 + P4 )2 · (P4 + P1 )2 Q4
141. Let the incircle ω of a triangle 4ABC touches its sides BC, CA, AB at the points D, E, F respectively. Now, let the line parallel to AB through E meets DF at Q, and the parallel to AB through D meets EF at T . Prove that the lines CF , DE, QT are concurrent. 142. ABCDEF is a hexagon whose opposite sides are parallel, this is, AB k DE, BC k EF and CD k F A. Show that triangles 4ACE and 4BDF have equal area. 143. Given a circle ω and a point A outside it. Construct a circle γ with centre A orthogonal to ω. 144. Prove that the circumcircles of the four triangles in a complete quadrilateral meet at a point. (Miquel Point) 145. Prove that the symmedian point of a triangle is the centroid of it’s pedal triangle with respect to that triangle. 146. Quadrilateral ABCD is convex with circumcircle (O), O lies inside ABCD. Its diagonals AC, BD intersect at S and let M , N , L, P be the orthogonal projections of S onto sides AB, BC, CD, DA. Prove that [ABCD] ≥ 2[M N LP ]. 147. Let ω be a circle in which AB and CD are parallel chords and ` is a line from C, that intersects AB in its midpoint L and ` ∩ ω = E. K is the midpoint of DE. Prove that KE is the angle bisector of ∠AKB. 148. Let ABC be an equilateral triangle and D, E be on the same side as C with the line AB, and BD is between BA, BE. Suppose ∠DBE = 90◦ , ∠EDB = 60◦ . Let F be the reflection of E about the point C. Prove that F A ⊥ AD. 149. In cyclic quadrilateral ABCD, AC · BD = 2 · AB · CD. E is the midpoint of AC. Prove that circumcircle of ADE is tangential to AB. 150. ABCD is a rhombus with ∠BAD = 60◦ . Arbitrary line ` through C cuts the extension of its sides AB, AD at M , N respectively. Prove that lines DM and BN meet on the circumcircle of 4BAD. 151. Let ABC be a triangle. Prove that there is a line(in the plane of ABC) such that the intersection of the interior of triangle ABC and interior of its reflection A0 B 0 C 0 has more than 2/3 the area of triagle ABC.
14
152. In triangle ABC, D, E, F are feet of perpendiculars from A, B, C to BC, AC, AB. Prove that the orthocenter of 4ABC is the incenter of 4DEF . 153. Let ABC be a triangle right-angled at A and ω be its circumcircle. Let ω1 be the circle touching the lines AB and AC, and the circle ω internally. Further, let ω2 be the circle touching the lines AB and AC and the circle ω externally. If r1 , r2 be the radii of ω1 , ω2 prove that r1 · r2 = 4A where A is the area of the triangle ABC. 154. The points D, E and F are chosen on the sides BC, AC and AB of triangle ABC, respectively. Prove that triangles ABC and DEF have the same centroid if and only if CE AF BD = = DC EA FB 155. Tangents to a circle form an external point A are drawn meeting the circle at B, C respectively. A line passing through A meets the circle at D, E respectively. F is a point on the circle such that BF is parallel to DE. Prove that F C bisects DE. 156. Let E be the intersection of the diagonals of the convex quadrilateral ABCD. Define [T ] to be the area of triangle T . If [ABE] + [CDE] = [BCE] + [DAE], prove that one of the diagonals bisect the other. 157. A line intersects AB, BC, CD, DA of quadrilateral ABCD in the points K, L, M , N . Prove that AK BL CM DN · · · =1 KB LC M D N A in magnitudes. 158. Let P Q be a chord of a circle. Let the midpoint of P Q be M . Let AB and CD be two chords passing through M . Let AC and BD meet P Q at H, K respectively. Prove that HA.HC KB.KD = 2 HM KM 2 159. Let ABCD be a trapezium with AB k CD. Prove that (AB 2 +AC 2 −BC 2 )(DB 2 +DC 2 −BC 2 ) = (BA2 +BD2 −AD2 )(CA2 +CD2 −AD2 ) 160. Given a rectangle ABCD and a point P on its boundary. Let S be the sum of the distances of P from AC and BD. Prove that S is constant as P varies on the boundary. 161. Let P and Q be two points on a semicircle whose diameter is XY (P nearer to X). Join XP and Y Q and let them meet at B. Let the tangents from P and Q meet at R. Prove that BR is perpendicular to XY . 15
162. Let a cyclic quadrilateral ABCD. L is the intersection of AC and BD and S = AD ∩ BC. Let M , N is midpoints of AB, CD. Prove that SL is a tangent of (M N L). 163. Let ABC be a right triangle with ∠A = 90◦ . Let D be such that CD ⊥ BC. Let O be the midpoint of BC. DO intersect AB at E. Prove that ∠ECB = ∠ADC + ∠ACD. 164. Given a circle ω and a point A outside it. A circle ω 0 passing through A is tangential to ω at B. The tangents to ω 0 at A, B intersect in M . Find the locus of M . 165. Triangle 4ABC has incircle (I) and circumcircle (O). The circle with center A and radius AI cuts (O) at X, Y . Show that line XY is tangent to (I). 166. Let ABCD be a cyclic quadrilateral with circumcircle ω. Let AB intersect DC at E. The tangent to ω at D intersect BC at F . The tangent to ω at C intersect AD at G. Prove that E, F , G are collinear. 167. Let ABC is a right triangle with C = 90◦ . H is the leg of the altitude from C, M is the mid-point of AB, P is a point in ABC such that AP = AC. Prove that P M is the bisector of ∠HP B if and only if A = 60◦ . 168. Two circles w1 and w2 meets at points P, Q. C is any point on w1 different from P, Q. CP meets w2 at point A. CQ meets w2 at point B. Find locus for ABC triangle’s circumcircle’s centres. 169. Consider a triangle 4ABC with incircle (I) touching its sides BC, CA, AB at A0 , B0 , C0 respectively. The triangle 4A0 B0 C0 is called the intouch triangle of 4ABC. Likewise, the triangle formed by the points of tangency of an excircle with the sidelines of 4ABC is called an extouch triangle. Let S0 , S1 , S2 , S3 denote the areas of the intouch triangle and the three extouch triangles respectively. Show that: 1 1 1 1 = + + S0 S1 S2 S3 170. Let ABCD be a convex quadrilateral such that ∠DAB = 90◦ and DA = DC. Let E be on CD such that EA ⊥ BD. Let F be on BD such that F C ⊥ DC. Prove that BC k F E. 171. (China TST 2007) Let ω be a circle with centre O. Let A, B be two points on its perimeter, and let CS and CT be two tangents drawn to ω from a point C outside the circle. Let M be the midpoint of the minor arc d M S and M T intersect AB in E, F respectively. The lines passing AB. through E, F perpendicular to AB cut OS, OT at X and Y respectively. Let ` be an arbitrary line cutting ω at the points P and Q respectively. Denote R = M P ∩ AB. If Z is the circumcentre of triangle P QR, prove that X, Y , Z are collinear. 16
172. Let ABCD be a convex quadrilateral such that ∠ABC = ∠ADC. Let E be the foot of perpendicular from A to BC and F is the foot of perpendicular from A to CD. Let M be the midpoint of BD. Prove that ME = MF. 173. Let H, K, I be the feet of the altitude from A, B, C of triangle ABC. Let M , N be the feet of the altitude from K, I of triangle AIK. Let P , Q be the point on HI, HK such that AP , AQ be perpendicular to HI, HK respectively. Prove that M , N , P , Q are collinear. 174. We have a 4ABC with ∠BAC = 90◦ . D is constructed such that AB = BD and A, B, D are three different collinear points. X is the foot of the altitude through A in 4ABC. Y is the midpoint of CX. Construct the circle τ with diametre CX. AC intersects τ again in F and AY intersects τ at G, H Prove that DX, CG, HF are concurrent. 175. Let ABCDE be a convex pentagon such that ∠EAB = 90◦ , EB = ED, AB = DC and AB k DC. Prove that ∠BED = 2∠CAB. 176. A straight line intersects the AB, BC internally and AC externally of triangles ABC in the points D, E, F respectively. Prove that the midpoints of AE, BF , CD are collinear. 177. Inside an acute triangle ABC is chosen point point K, such that ∠AKC = 2 AB AK = .. where A1 and C1 are the midpoints of BC 2∠ABC and KC BC and AB. Prove, that K lies on circumcircle of triangle A1 BC1 . 178. M is the midpoint of the side BC of 4ABC and AC = AM +AB. Incircle (I) of 4ABC cuts A-median AM at X, Y . Show that ∠XIY = 120◦ . 179. Let ABC be an isosceles triangle with AB = AC. Let P , Q be points on the side BC such that ∠AP C = 2∠AQB. Prove that BP = AP + QC. 180. Let BC be a diameter of the circle O and let A be an interior point. Suppose that BA and CA intersect the circle O at D and E, respectively. If the tangents to the circle O at E and D intersect at the point M , prove that AM is perpendicular to BC. 181. Let ABC be triangle and G its centroid. Then for any point M , we have M A2 + M B 2 + M C 2 = 3M G2 + GA2 + GB 2 + GC 2 . 182. Given two non-intersecting and non-overlapping circles and a point A lying outside the circles. Prove that there are exactly four circles(straight lines are also considered as circles) touching the given two circles and passing through A.
17
183. A non-isosceles triangle ABC is given. The altitude from B meets AC at E. The line through E perpendicular to the B-median meets AB at F and BC at G. Prove that EF = EG if, and only if, ∠ABC = 90◦ 184. Given a triangle ABC and a point T on the plane whose projections on AB, AC are C1 , B1 respectively. B2 is on BT such that AB2 is perpendicular to BT and C2 is on CT such that AC2 is perpendicular to CT . Prove that B1 B2 and C1 C2 intersect on BC. 185. Let ABCD be a cyclic quadrilateral with ∠BAD = 60◦ . Suppose BA = BC + CD. Prove that either ∠ABD = ∠CBD or ∠ABC = 60◦ . 186. In a quadrilateral ABCD we have AB k CD and AB = 2 · CD. A line ` is perpendicular to CD and contains the point C. The circle with centre D and radius DA intersects the line ` at points P and Q. Prove that AP ⊥ BQ. 187. In triangle ABC, a circle passes through A and B and is tangent to BC. Also, a circle that passes through B and C is tangent to AB. These two circles intersect at a point K other than B. If O is the circumcenter of ABC, prove that ∠BKO = 90◦ . 188. Four points P, Q, R, S are taken on the sides AB, BC, CD, DA of a quadrilateral such that AP BQ CR DS · · · =1 P B QC RD SA Prove that P Q and RS intersect on AC. 189. Let D be the midpoint of BC of triangle ABC. Let its incenter be I and AI intersects BC at E. Let the excircle opposite A touches the side BC at F . Let M be the midpoint of AF . Prove that AD, F I, EM are concurrent. 190. 4ABC is scalene and its B− and C− excircles (Ib ) and (Ic ) are tangent to sideline BC at U , V . M is the midpoint of BC and P is its orthogonal projection onto line Ib Ic . Prove that A, U , V , P are concyclic. 191. Let H be the orthocenter of acute 4ABC. Let D, E, F be feet of perpendiculars from A, B, C onto BC, CA, AB respectively. Suppose the squares constructed outside the triangle on the sides BC, CA, AB has area Sa , Sb , Sc respectively. Prove that Sa + Sb + Sc = 2(AH · AD + BH · BE + CH · CF ) 192. In rectangle ABCD, E is the midpoint of BC and F is the midpoint of AD. G is a point on AB (extended if necessary); GF and BD meet at H. Prove that EF is the bisector of angle GEH. 18
193. P is a point in the minor arc BC of the circumcircle of a square ABCD, prove that PA + PC PD = PB + PD PA 194. ABCD is a cyclic trapezoid with AB k CD. M is the midpoint of CD and AM cuts the circumcircle of ABCD again at E. N is the midpoint of BE. Show that N E bisects ∠CN D. 195. A line is drawn passing though the centroid of a 4ABC meeting AB and AC at M and N respectively. Prove that AM · N C + AN · M B = AM · AN 196. Let the isosceles triangle ABC where AB = AC. The point D belongs to the side BC and the point E belongs to AC. C = 50◦ , ∠ABD = 80◦ and ∠ABE = 30◦ , find ∠BED. 197. Let S be the area of 4ABC and BC = a. Let r be its inradius and ra be its exradius opposite A. Prove that S=
arra ra − r
198. A line segment AB is divided by internal points K, L such that AL2 = AK · AB. A circle with centre A and radius AL is drawn. For any point P on the circle, prove that P L bisects ∠KP B. 199. Let 4ABC be a triangle with ∠A = 60◦ . Let BE and CF be the internal angle bisectors of ∠B and ∠C with E on AC and F on AB. Let M be the reflection of A in the line EF . Prove that M lies on BC. (Regional Olympiad 2010, India) 200. In triangle ABC, Z is a point on the base BC. Lines passing though B and C that are parallel to AZ meet AC and AB at X, Y respectively. Prove that: 1 1 1 + = BX CY AZ 201. Let ABCD be a trapezoid such that AB > CD, AB k CD. Points K AK and L lie on the segments AB and CD respectively such that KB = DL LC . Suppose that there are points P and Q on the segment KL satisfying ∠AP B = ∠BCD and ∠CQD = ∠ABC. Prove that P , Q, B, C are concyclic. 202. I is the incenter of 4ABC. Let E be on the extension of CA such that CE = CB + BA and F is on the extension of BA such that BF = BC + CA. If AD is the diameter of the circumcircle of 4ABC, prove that DI ⊥ EF . 19
203. ABCD is a parallelogram with diagonals AC, BD. Circle Γ with diameter AC cuts DB at P , Q and tangent line to Γ through C cuts AB, AD at X, Y . Prove that points P , Q, X, Y are concyclic. 204. Two triangles have a common inscribed in and circumscribed circle. Sides of one of them relate to the inscribed circle at the points K, L and M , sides of another triangle at points K1 , L1 and M1 . Prove that orthocentres of traingles KLM and K1 L1 M1 are match. 205. ABCD is a convex quadrilateral with ∠BAD = ∠DCB = 90◦ . Let X and Y be the reflections of A and B about BD and AC respectively. P ≡ XC ∩ BD and Q ≡ DY ∩ CA. Show that AC ⊥ P Q. 206. In triangle ABC, ∠A = 2∠B = 4∠C. Prove that 1 1 1 = + AB BC AC 207. Point P lies inside 4ABC such that ∠P BC = 70◦ , ∠P CB = 40◦ , ∠P BA = 10◦ and ∠P CA = 20◦ . Show that AP ⊥ BC. 208. The sides of a triangle are positive integers such that the greatest common divisor of any 2 sides is 1. Prove that no angle is twice of another angle in the triangle. 209. Two circles with centres A, B intersect on points M , N . Radii AP and BQ are parallel(on opposite sides of AB). If the common external tangents meet AB at D and P Q meet AB at C, prove that ∠CN D is a right angle. 210. In an acute triangle 4ABC, the tangents to its circumcircle at A and C intersect at D, the tangents to its circumcircle at C and B and intersect at E. AC and BD meet at R while AE and BC meet at P . Let Q and S be the mid-points of AP and BR respectively. Prove that ∠ABQ = ∠BAS. 211. Two circles Γ1 and Γ2 meet at P , Q. Their common external tangent (closer to Q) touches Γ1 and Γ2 at A, B. Line P Q cuts AB at R and the perpendicular to P Q through Q cuts AB at C. CP cuts Γ1 again at D and the parallel to AD through B cuts CP at E. Show that RE ⊥ CD. 212. Let ABCD be a convex quadrilateral such that the angle bisectors of ∠DAB and ∠ADC intersect at E on BC. Let F be on AD such that ∠F ED = 90◦ − ∠DAE. If ∠F BE = ∠F DE, prove that EB 2 + EF · ED = EB(EF + ED) 213. Let ABC be a triangle. Let P be a point inside such that ∠BP C = CP CB 180◦ − ∠ABC and P B = BA . Prove that ∠AP B = ∠CP B.
20
214. Let ABCD be a cyclic quadrilateral, and let rXY Z denote the inradius of 4XY Z. Prove that rABC + rCDA = rBCD + rDAB 215. 4ABC is right-angled at A. H is the projection of A onto BC and I1 , I2 are the incenters of 4AHB and 4AHC. Circumcircles of 4ABC and 4AI1 I2 intersect at A, P . Show that AP , BC, I1 I2 concur. 216. An ant is crawling on the inside of a cube with side length 6. What is the shortest distance it has to travel to get from one corner to the opposite corner? 217. If Ia is the excenter opposite to side A and O is the circumcenter of 4ABC. Then prove that: (OIa )2 = R2 + 2Rra 218. The two circles below have equal radii of 4 units each and the distance between their centers is 6 units. Find the area of the region formed by common points. 219. Triangle ABC and its mirror reflection A0 B 0 C 0 are arbitrarily placed on a plane. Prove that the midpoints of the segments AA0 , BB 0 and CC 0 lie on the same straight line. 220. The convex hexagon ABCDEF is such that ∠BCA = ∠DEC = ∠F AE = ∠AF B = ∠CBD = ∠EDF Prove that AB = CD = EF . √ 221. Let ABC be a triangle such that BC = 2AC. Let the line perpendicular to AB passing through C intersect the perpendicular bisector of BC at D. Prove that DA ⊥ AC. 222. Three circles with centres A, B, C touch each other mutually, say at points X, Y , Z. Tangents drawn at these points are concurrent (no need to prove that) at point P such that P X = 4. Find the ratio of the product of radii to the sum of radii. 223. Hexagon ABCDEF is inscribed in a circle of radius R centered at O; let AB = CD = EF = R. Prove that the intersection points, other than O, of the pairs of circles circumscribed about 4BOC, 4DOE and 4F OA are the vertices of an equilateral triangle with side R. 224. Triangle ABC has circumcenter O and orthocenter H. Points E and F are chosen on the sides AC and AB such that AE = AO and AF = AH. Prove that EF = OA.
21
225. Let AD, BE, CF be the altitudes of triangle ABC. Show that the triangle whose vertices are the orthocenters of triangles AEF , BDF , CDE is congruent to triangle DEF . 226. Suppose `1 and `2 are parallel lines and that the circle Γ touches both `1 and `2 , the circle Γ1 touches `1 and Γ externally in A and B, respectively. Circle Γ2 touches `2 in C, Γ externally in D and Γ1 externally at E. Prove that AD and BC intersect in the circumcenter of triangle BDE. 227. 4ABC is scalene and M is the midpoint of BC. Circle ω with diameter AM cuts AC, AB at D, E. Tangents to ω at D, E meet at T . Prove that T B = T C. 228. Point P lies inside triangle ABC and ∠ABP = ∠ACP . On straight lines AB and AC, points C1 and B1 are taken so that BC1 : CB1 = CP : BP . Prove that one of the diagonals of the parallelogram whose two sides lie on lines BP and CP and two other sides (or their extensions) pass through B1 and C1 is parallel to BC. 229. Let ABC be a right angled triangle at A. D is a point on CB. Let M be the midpoint of AD. CM intersects the perpendicular bisector of AB at E. Prove that BE k DA. 230. Prove that the pedal triangle of the Nine-point centre of a triangle with angles 75◦ , 75◦ , 30◦ has to be equilateral. 231. 4ABC is right-angled at A. D and E are the feet of the A-altitude and A-angle bisector. I1 , I2 are the incenters of 4ADB and 4ADC. Inner angle √ bisector of ∠DAE cuts BC and I1 I2 at K, P . Prove that P K : P A = 2 − 1. 232. In acute triangle ABC, there exists points D and E on sides AC, AB respectively satisfying ∠ADE = ∠ABC. Let the angle bisector of ∠A hit BC at K. P and L are projections of K and A to DE, respectively, and Q is the midpoint of AL. If the incenter of 4ABC lies on the circumcircle of 4ADE, prove that P , Q, and the incenter of 4ADE are collinear. 233. Let (O) is the circumcircle ABC. D, E lies on (BC). (U ) touches to AD, BD at M and intouches (O). (V ) touches to AE, BE at N and intouches (O). d touches external to (U ) and (V ). P lie on d and d touches to the circumcircle of BP C. A circle touches to d at P and BC at H. Prove \ P H is the bisector of M P N . (BC) be circle with diameter BC. 234. In triangle ABC, the median through vertex I is mi , and the height through vertex I is hi , for I ∈ A, B, C. Prove that if 2 ma 2 mb 2 mc hb hc ha =1 hb hc hc ha ha hb then ABC is equilateral. 22
235. Let D and E are points on sides AB and AC of a 4ABC such that DE k BC, and P is a point in the interior of 4ADE, P B and P C meet DE at F and G respectively. Let O and O0 be the circumcenters of 4P DG and 4P F E respectively. Prove that AP ⊥ OO0 . 236. Let ABCD be a parallelogram. If E ∈ AB and F ∈ CD, and provided that AF ∩ DE = X, BF ∩ CE = Y , XY ∩ AD = L, XY ∩ BC = M ; show that AL = CM . 237. In a triangle ABC, P is a point such that angle ∠P BA = ∠P CA. Let B 0 , C 0 be the feet of perpendiculars from P onto AB and AC. If M is the midpoint of BC, the prove that M lies on the perpendicular bisector of B 0 C 0 . 238. The lines joining the three vertices of triangle ABC to a point in its plane cut the sides opposite verticea A, B, C in the points K, L, M respectively. A line through M parallel to KL cuts BC at V and AK at W . Prove that V M = M W . 239. Let ABCD be a parallelogram. Let M ∈ AB, N ∈ BC and denote by P , Q, R the midpoints of DM , M N , N D, respectively. Show that the lines AP , BQ, CR are concurrent. 240. Let (O1), (O2) touch the circle (O) internally at M , N . The internal common tangent of (O1 ) and (O2 ) cut (O) at E, F , R, S. The external common tangent of (O1 ), (O2 ) cut (O) at A, B. Prove that AB k EF or AB k SR. 241. Let H be the orthocenter of the triangle ABC. For a point L, denote the points M , N , P are chosen on BC, CA, AB, respectively, such that HM , HN , HP are perpendicular to AL, BL, CL, respectively. Prove that M , N , P are collinear and HL is perpendicular to M P . 242. The bisector of each angle of a triangle intersects the opposite side at a point equidistant from the midpoints of the other two sides of the triangle. Find all such triangles. 243. ABCD trapezoid’s bases are AB, CD with CD = 2 · AB. There are P , Q BQ DP = 2; = 3 : 4. Find ratio of ABQP , points on AD, BC sides and PA QC CDP Q quadrilaterals areas. 244. In convex quadrilateral ABCD we found two points K and L, lying on segments AB and BC, respectively, such that ∠ADK = ∠CDL. Segments AL and CK intersects in P . Prove, that ∠ADP = ∠BDC. 245. Let ABCD be a parallelogram and P is a point inside such that ∠P AB = ∠P CB. Prove that ∠P BC = ∠P DC.
23
246. Consider a triangle ABC and let M be the midpoint of the side BC. Suppose ∠M AC = ∠ABC and ∠BAM = 105◦ . Find the measure of ∠ABC. 247. Let AA1 , BB1 , CC1 be the altitudes of acute angled triangle ABC; OA , OB , OB are the incenters of triangles AB1 C1 , BC1 A1 , CA1 B1 , respectively; TA , TB , TC are the points of tangent of incircle of triangle ABC with sides BC, CA, AB respectively. Prove, that all sides of hexagon TA OC TB OA TC OB are equal. 248. Let ABC be a triangle and P is a point inside. Let AP intersect BC at D. The line through D parallel to BP intersects the circumcircle of 4ADC at E. The line through D parallel to CP intersects the circumcircle of 4ADB at F . Let X be a point on DE and Y is a point on DF such that ∠DCX = ∠BP D and ∠DBY = ∠CP D. Prove that XY k EF . 249. Prove that if N ∗ , O is the isogonal conjugate of the nine-point centre of 4ABC and the circumcentre of 4ABC respectively, then A, N ∗ , M are collinear, where M is the circumcentre of 4BOC. 250. So here’s easy one in using vectors. ABCDE is convex pentagon with S area. Let a, b, c, d, e are area of 4ABC, 4BCD, 4CDE, 4DEA, 4EAB. Prove that: S 2 − S(a + b + c + d + e) + ab + bc + cd + de + ea = 0 251. ABC is a triangle with circumcentre O and orthocentre H. Ha , Hb , Hc are the foot of the altitudes from A, B, C respectively. A1 , A2 , A3 are the circumcentres of the triangles BOC, COA, AOB respectively. Prove that Ha A1 , Hb A2 , Hc A3 concurr on the Euler’s line of triangle ABC. 252. The incircle (I) of a given scalene triangle ABC touches its sides BC, CA, AB at A1 , B1 , C1 , respectively. Denote ωB , ωC the incircles of quadrilaterals BA1 IC1 and CA1 IB1 , respectively. Prove that the internal common tangent of ωB and ωC different from IA1 passes through A. 253. Let ω1 , ω2 be 2 circles externally tangent to a circle ω at A, B respectively. Prove that AB and the common external tangents of ω1 , ω2 are concurrent. 254. Let AC and BD be two chords of a circle ω that intersect at P . A smaller circle ω1 is tangent to ω at T and AP and DP at E, F respectively. (Note that the circle ω1 will lie on the same side of A, D with respect to P .) ˆ of ω, and if I is the incentre of ACD, show Prove that T E bisects ABC that F = ω1 ∩ EI =⇒ DF is tangent to ω1 . 255. Assume that the point H is the orthocenter of the given triangle ABC and P is an arbitrary point on the circumcircle of ABC. E is a point on AC such that BE ⊥ AC. Let us construct to parallelograms P AQB and P ARC. Assume that AQ and HR intersect at point X. Prove that EX k AP . 24
256. Let AD, BE be the altitudes of triangle ABC and let H be the orthocenter. The bisector of the angle DHC meets the bisector of the angle B at S and meet AB, BC at P , Q, respectively. And the bisector of the angle B meets the line M H at R, where M is the midpoint of AC. Show that RP BQ is cyclic. 257. Prove that the Simson lines of diametrically opposite points on circumcircle of triangle ABC intersect at nine point circle of the triangle. 258. In an equilateral triangle ABC. Prove that lines trough A that trisects outward semicircle on BC as diameter trisect BC as well. 259. Prove that the feet of the four perpendiculars dropped from a vertex of a triangle upon the four bisectors of the two other angles(two internal and two external angle bisectors) are collinear. 260. Let ABC be a triangle. Let the angle bisector of ∠A, ∠B intersect BC, AC at D, E respectively. Let J be the incenter of 4ACD. Suppose that EJDB is cyclic. Prove that ∠CAB is equal to either ∠CBA or 2∠ACB. 261. (O) is a circle and P is a point outside (O). Tangents from P to (O) touch (O) at A, B. M , N are the midpoints of P A, P B and Q is a point on M N . Tangents from Q to (O) touch (O) at C, D. E ≡ AB ∩ CDandF ≡ CD ∩ M N . Show that OE k P F . 262. Let H be the orthocentre of a triangle ABC. X is an arbitrary point in the plane of ABC such that the circle with diameter XH again meets AH, BH, CH at points A1 , B1 , C1 ; and the lines AX, BX, CX at A2 , B2 ,C2 respectively. Prove that the lines A1 A2 , B1 B2 and C1 C2 concur. 263. A semicircular piece of paper with radius 2 is creased and folded along a chord so that the arc is tangent to the diameter AB at C. If C divides AB in the ratio 3:1, determine the length of the crease. 264. Show that two triangles with equal perimeter, equal area and one equal angle are congruent. 265. ABCD is a tetrahedron with AB = DC and BC = AD. M is the midpoint of AC and N is the midpoint of BD. Prove that M N is perpendicular to both AC and BD. 266. Let AA0 , BB 0 , CC 0 be concurrent cevians of a triangle ABC. Let BC ∩ B 0 C 0 = P , CA ∩ C 0 A0 = Q, AB ∩ A0 B 0 = R. Prove that P , Q, R are collinear. Let P , Q, R be a line `. Let p, q, r be the cevians drawn from A, B, C to BC, CA, AB parallel to B 0 C 0 , C 0 A0 , A0 B 0 respectively. Prove that ` bisects p, q, r. 267. Let ABCD be a cyclic quadrilateral. E is an arbitrary point on AB, and F an arbitrary point on DC. AF intersects the circle at M and DE intersects at N . Prove that BC, EF and M N are either parallel or concurrent. 25
268. If A, B, C, D are four points in space, prove that AB + BC + CD + DA ≥ AC + BD. 269. Let ABCD be a cyclic quadrilateral, with opposite sides not parallel. Let E be the fourth point of the parallelogram with the other three vertices being A, B, C. Let AD, BC intersect at M , and AB, DC intersect at N , and EC, M N intersect at F . Prove that DEN F is a cyclic quadrilateral. 270. A is a point inside a semicircle (M ) with diameter BC. The incircle (I) of 4ABC touches AC, AB at E, F and EF cuts (M ) again at X, Y . Show that ∠BAC equals the measure of the arc XY of (M ). 271. H is the orthocentre of triangle ABC. The circle with diameter AH meets the circumcircle of ABC again at P . Prove that P H bisects BC. 272. Prove that a straight line dividing the perimeter and the area of a triangle in the same ratio passes through its incenter. 273. Let the internal angle bisectors of ∠ABH and ∠ACH meet at some point X, where H is the orthocentre of 4ABC. Denote, by K, L the midpoints of AH, BC, respectively. Show that X lies on KL. 274. A circle is tangent to BC of equilateral ABC at D (D between B and C). It cuts AB internally at M and N and AC internally at P and Q. Show that BD + AM + AN = CD + AP + AQ. 275. Consider 4 points in a plane no three of them are collinear. Prove that the four pedal circles each of which corresponds to one of the points under consideration with respect to the triangle formed by the other three have a common point. 276. In a triangle 4ABC with incenter I, let D and E be the feet of the A- and B- angle bisectors. M is the midpoint of ED. Show that IM ⊥ AB ⇐⇒ either CA = CB or ∠ACB = 90◦ . 277. Suppose, O is the circumcenter of a triangle ABC. A line ` passing through O intersects AB, BC, CA at C 0 , A0 , B 0 respectively. A circle centered at O intersects AO, BO, CO at A1 , B1 , C1 respectively. Prove that the circles A1 OA0 , B1 OB 0 , C1 OC 0 have a common point other than O. 278. Let ABC be a triangle with orthocentre H. The point D is diametrically opposite to A with respect to (ABC). A line passing through D meets AB, AC at E, F . The feet of perpendiculars from E, F onto AC, AB are J, K. Show that the points J, K and H are collinear. 279. Let 4ABC be an acute triangle, H the altitude from A, and D, E, Q the feet of the perpendiculars from an arbitrary point P in the triangle onto AB, AC, AH, respectively. Prove that: |AB · AD − AC · AE| = BC · P Q 26
280. Let (O) is circumcircle of triangle XY Z. G is centroid of XY Z. I, H, K are circumcircles of triangle GY Z, GZX, GXY . Prove O is the centroid of IHK. 281. On a circle ω, five points A, B, C, D, E are randomly chosen in any order. Prove that the simson lines of D, E wrt 4ABC meet on the orthopole of the line DE wrt 4ABC. 282. D, E, F are the midpoints of the sides BC, CA, AB of 4ABC. AD, BE, CF cut the circumcircle of 4ABC again at M , N , L. Show that 4M N L is isosceles with apex M ⇐⇒ either b = c or 2a2 = b2 + c2 . 283. Suppose, in an equilateral triangle ∆ABC a point P is chosen inside the triangle. Find the minimum upper-bound of P A · P B · P C where ∆ABC is inscribed in a circle of radius 1. 284. From point P outside a circle (O) tangents P A and P B are drawn. From a point Q outside (O) and on the line through A, B tangents QC and QD are drawn. Prove that P , C, D are collinear. 285. In triangle ABC points D, E, F are on sides BC, AC, AB, respectively such that AD, BE, CF are concurrent in point M . Areas of triangles M F A, M DB, M EC are equal and also their perimeter are equal. Prove that triangle ABC is equilateral. 286. Given ABCD is a convex and AC intersect BD at M . Let R1 , R2 , R3 , R4 are inradius of triangles M AB, M BC, M CD, M DA respectively such that R11 + R13 = R12 + R14 . Prove that AB + CD = AD + CB. 287. Let O be the circumcenter of non-isosceles 4ABC. Draw a circle passing through B, C and tangent to AC. Draw another circle passing through A, C and tangent to BC. Let them intersect at a point P different from C. Prove that OP ⊥ P C. 288. Let ABC be a triangle and M , X and Y points in AB, AC and BC, respectively, such that AX = M X and BY = M Y . Let O be the circumcenter of triangle ABC. Show that cuadrilateral CXOY is cyclic. 289. Let ABCD be a cyclic quadrilateral. Lines AC and BD intersect at E. Segments AC and BD have midpoints M , N respectively. Let lines AD and BC intersect at F . Let the orthocentre of 4F AB be H. Prove that HE ⊥ M N . 290. The three squares ACC1 A00 , ABB1 A0 , BCDE are constructed externally on the sides of a triangle ABC. Let P be the centre of BCDE. Prove that A0 C, A00 B, P A concur. 291. Let a be the side length and b the diagonal length of a regular pentagon. Prove that ab − ab = 1.
27
292. The incircle of triangle ABC is tangent to sides AC and BC at M and N , respectively. The bisectors of the angles at A and B intersect M N at points P and Q, respectively. Let I be the incenter of triangle ABC. Prove that M P · IA = BC · IQ. 293. M is an arbitrary point on the circle (O) and P is a point outside (O). Circle ω with diameter P M meets (O) again at N . Tangent line to ω through P cuts M N at Q. The perpendicular to OP through Q cuts P M at A. AN cuts (O) again at B and BM cuts OP at C. Prove that AC ⊥ OQ. 294. Take two squares ABCD and P QRS in the plane. Prove that the quadrangle formed by the midpoints of AP , BQ, CR, DS is itself a square. 295. D is a point on the minnor arc AC of the circumcircle of triangle ABC with center O. Take point P on segment AB such that ∠ADP = ∠OBC and take a point Q on segment BC such that ∠CDQ = ∠OBA. Prove that: ∠DP Q = ∠DOC. 296. In 4ABC, let X, Y , Z be the midpoints of BC, CA, AB respectively, and let P , Q, R be the midpoints of AX, BY , CZ respectively. Find and prove the invariant value of AQ2 + AR2 + BP 2 + BR2 + CP 2 + CQ2 . AB 2 + BC 2 + CA2 297. Let S be a point in triangle ABC such that ∠BAS = ∠CBS = ∠ACS = k. Prove cot k = cot a + cot b + cot c. 298. Let AD, BE, CF be the altitudes of a triangle ABC. The line through D, parallel to EF meets AB at R and AC at Q. Let P = EF ∩ CB. Prove that (P QR) passes through the midpoint M of side BC. 299. Let ABC be a triangle and H the intersection of the perpendicular from C to AB. Let F be the midpoint of CH and E the midpoint of AB. Points Q and P are on sides AC and BC respectively such that P Q is parallel to AB. Let R be the intersection of the perpendicular drawn from Q to AB. Let S be the intersection of EF and P R. Show that S is the midpoint of P R. 300. Let Γ be the circumcircle of 4ABC with center O. E is the excenter opposite A. Draw a line ` through E perpendicular to AE. Let X, Y be points on ` such that ∠XAO = ∠Y AO = ∠BAE. Prove that the incenter of 4AXY lies on Γ. 301. The incircle of 4ABC touches its sides BC, CA, AB at D, E, F . Show that the area of 4DEF equals the harmonic mean between the areas of 4BEF , 4CEF .
28
302. Let ABC be a triangle with internal angle bisectors AD, BE, CF such that D ∈ BC, E ∈ CA, F ∈ AB. Show that if ∠BAC = 120◦ , then the circle with EF as diameter passes through D. 303. Let Γ1 , Γ2 , Γ3 be 3 intersecting circles. Γ1 intersects Γ2 at A, D. Γ2 intersects Γ3 at B, E. Γ3 intersects Γ1 at C, F . Let X, Y , Z be the intersections of BC, EF ; AC, DF ; AB, DE. Prove that X, Y , Z are collinear. 304. Let A, B, C be three collinear points, with B between A and C. Equilateral triangles ABD, BCE, CAF are constructed with D, E on one side of the line AC and F on the opposite side. Prove that the centroids of the triangles are the vertices of an equilateral triangle. Prove that the centroid of this triangle lies on the line AC. 305. Determine whether there exist 1976 nonsimilar triangles with angles α, β, γ, each of them satisfying the relations 12 12 sin α + sin β + sin γ = and sin α sin β sin γ = cos α + cos β + cos γ 7 25 306. S1 and S2 are two circles touching each other externally and S is the circle touching S1 and S2 internally. A direct common tangent to S1 and S2 intersects S at B and C and the common tangent of S1 and S2 at P intersects S at A so that A and P are on the same side of BC. Prove that P is the in-center of 4ABC. 307. Let ABCDEF GHI be a regular nonagon with circumcenter O. Let M , N , P be midpoints of arc AB, segment BC and segment OM respectively. Find ∠ON P . 308. ??? 309. Let ABCDEF GHI be a regular nonagon again. Prove that AE = AB + AC 310. Let ABCD be the convex quadrilaterals. Show that [ABCD] ≤
1 (|AC|2 + |BD|2 ) 4
where [ABCD] is area of ABCD. 311. Let ABCD be a cyclic quadrilateral. The lines AB and CD meet at P , and the lines AD and BC meet at Q. Let E and F be the points where the tangents from Q meet the circumcircle of ABCD. Prove that P , E, F are collinear.
29
312. Let S1 be a semicircle with centre O and diameter AB. A circle C1 with centre P is drawn, tangent to S1 , and tangent to AB at O. A semicircle S2 is drawn, with centre Q on AB, tangent to S1 and to C1 . A circle C2 with centre R is drawn, internally tangent to S1 and externally tangent to S2 and C1 . Prove that OP RQ is a rectangle. 313. Let P ∈ BC and Q ∈ DC, where ABCD is a square. Assume ^P AQ = Show that BP + DQ = P Q.
π 4.
314. Let H be the orthocenter of triangle ABC. Prove that the circumcircles of triangles HAB, HBC, and HCA have equal radii. 315. Let ABCD be a convex quadrilateral inscribed in a circle. Let M be the intersection point of its diagonals. Let E, F , G, H be the feet of the perpendiculars from M to AB, BC, CD, DA respectively. Determine, with proof, the incenter of EF GH. 316. If P is the Fermat point of the triangle ABC then prove that the Euler lines of the triangles P AB, P BC and P CA are concurrent and the point of concurrence is G (G is the centroid of 4ABC). 317. Equilateral triangles ABC 0 and AB 0 C are built outwardly on the sides of non-isosceles triangle ABC, and equilateral triangles ABC 00 and AB 00 C are built inwardly on the sides of triangle ABC. Let the intersection of BB 0 and CC 00 be D and the intersection of BB 00 and CC 0 be E. Prove that triangle ADE is equilateral. 318. Let D, E, F be the feet of perpendiculars from A, B, C onto their opposite sides in 4ABC, with circumcenter O and orthocenter H. Suppose DE intersects AB at X, and DF intersects AC at Y . Prove that XY ⊥ OH. 319. Let I be the incenter of triangle ABC and A0 , B 0 and C 0 the points in which the incircle is tangent to the sides BC, AC and AB, respectively. Let P be the intersection of AA0 with BB 0 , M the intersection of AC with A0 C 0 and N the intersection of BC with B 0 C 0 . Show that IP is perpendicular to M N . 320. Let ABC be a triangle, and D is any point on BC. Draw the incircle of 4ABD with center I. The incircle touches the sides BC, AD at P , Q. Draw the incircle of 4ACD with center J. The incircle touches the sides BC, AD at R, S. Prove that P Q, RS, IJ are concurrent. 321. The circumferences of three circles of same radius r intersect at a common point. Determine the minimum and maximum values of the radius of the circle through each of the three pairwise intersections of the three circles, in terms of r. 322. In a triangle the pedal triangle of its nine-point center is constructed. In the pedal triangle the again the pedal triangle of its nine-point center is
30
drawn. Continuing in this way the triangle will tend to an equilateral triangle after infinite steps. 323. ABC is a triangle and P is any point inside it. X, Y are the feet of perpendiculars from P to AB, AC respectively. Z, W are the feet of the perpendiculars from A to BP , CP respectively. Prove that the lines ZY , W X and BC are concurrent. 324. Show that the internal bisector of ∠A does not exceed ABC.
AB+AC 2
in triangle
325. Use the fact that the A-angle bisector in triangle ABC has length A 2bc b+c cos 2 and similar for B, C to prove that if two angle bisectors in a triangle are equal, then it is isosceles. 326. In a triangle ABC let A1 , B1 , C1 be the diametrically opposite points of the the feet of the altitudes of the w.r.t the ninepoint circle. Prove that the lines AA1 , BB1 and CC1 are either concurrent or parallel to each other. 327. Consider a square ABCD and let P and Q be the points on the sides AB and BC such that P B = BQ. Let H be the foot of perpendicular dropped from B to P C. Prove that ∠DHQ = 90◦ . 328. Let A0 B 0 C 0 be the contact triangle of ABC. A1 , B1 , C1 are the reflections of A0 , B 0 , C 0 on AI, BI, CI where I is the incenter of ABC. Prove that A2 A1 , B2 B1 , C2 C1 are concurrent where A2 , B2 , C2 are the midpoints of BC, CA, AB. 329. Let 4A1 B1 C1 be the in-touch triangle of 4ABC. Prove that the Euler line of 4A1 B1 C1 passes through the circumcenter of 4ABC. 330. Find Fermat distance pure geometrically in terms of side lengths and area of the triangle. (Fermat distance is AF + BF + CF where F is the Fermat point of ∆ABC). 331. Let H be the orthocenter of triangle ABC, let O be the circumcenter and M be the midpoint of BC. Prove that 12 AH = OM . 332. The tangents at B, C to circle ABC meet at T . Let L, M , N be the midpoints of BC, AC, AB respectively. The lines AT and N L meet at X; the lines AT and LM meet at Y . Prove that BX is parallel to CY . 333. ABCD is a parallelogram. A line through A intersects BC at E and CD at F . Prove that the triangles BEF and DCE have the same area. 334. Prove geometrically(i.e without any calculation) that R ≥ 2r where R, r means usual notation.
31
335. Let [ABC] = S1 and [ABH] = S2 , where H is the orthocenter of triangle ◦ ABC. Let K be √ the point on line CH such that ∠AKB = 90 . Show that [ABK] = S1 S2 . 336. S is a circle, and C1 , C2 are circles with radius r1 , r2 internally tangent to and inside S, and C1 , C2 are tangent to each other. Circle C has radius r, internally tangent to S and externally tangent to C1 , C2 . Prove that 1 1 1 √ <√ +√ r1 r2 r 337. In a triangle 4ABC consider any line ` passing through the foot of the perpendicular from A on BC. Take points E, F on ` such that ∠BEA = 90◦ , ∠AF C = 90◦ . Midpoint of EF , BC be M , N respectively. Prove that AM ⊥ M N . 338. Given triangle ABC (AB < AC) inscribed in (O) whose diameter is AD. Tangent of (O) at D cuts BC at S. SO cuts AB, AC at E, F respectively. Prove that OE = OF . 339. In a triangle 4ABC, isosceles triangles 4A0 BC, 4B 0 CA, 4C 0 AB with same vertical angle are drawn on outside of 4ABC. Prove that AA0 , BB 0 , CC 0 are concurrent. 340. Let p be the inradius of the pedal(orthic) triangle. R the circumradius of 4ABC, and q the circumradius of the tangential triangle, then show that R2 = 2pq. 341. The circle γ1 centered at O1 intersects the circle γ2 centered at O2 at two points P and Q. The tangent to γ2 at P intersects γ1 at point A and the tangent to γ1 at P intersects γ2 at point B where A and B are distinct from P . Suppose P Q · O1 O2 = P O1 · P O2 and ∠AP B is acute, determine ∠AP B in degree. 342. Let S be the symmedian point of 4ABC. D, E, F are feet of perpendiculars from S onto BC, CA, AB. Prove that S is the centroid of 4DEF . 343. In a triangle ABC, I is its incenter. Let A1 , B1 , C1 are arbitrary points on the segments AI, BI and CI respectively. The perpendicular bisector of the segments AA1 , BB1 and CC1 meet at the points A2 , B2 and C2 respectively. Prove that the circumcinter of the triangles ABC and A2 B2 C2 coincide iff I is the orthocenter of the triangle A1 B1 C1 . 344. Given any four non-collinear coplanar points A, B, C, D, prove that the nine-point circles of ABC, BCD, CDA, DAB pass through a common point.
32
345. Let ABCD be a convex quadrilateral. Prove that we can choose 3 of the vertices, say A, B, C and draw a parallelogram ABCE, which contains the point D. (the point D may lie on an edge or vertex). 346. Let ABCD be a cyclic quadrilateral. Let P , Q, R be the feet of the perpendiculars from D to the lines BC, CA, AB respectively. Show that P Q = QR iff the angle bisectors of ∠ABC and ∠ADC concur at the line AC. 347. Let ABCD be a convex quadrilateral. M , N are midpoints of AC, BD respectively. Prove that AB + BC + CD + DA ≥ AC + BD + 2M N 348. A, B, C are three collinear points and O is a point not on the line ABC. Prove that O and the circumcenters of ABO, BCO, CAO are concyclic. 349. Three congruent circles have a common point O. The circles (taken in pairs) also meet at A, B, C. Let XY Z be the triangle containing these three circles, such that every side of the triangle is tangent to exactly two of the given circles. Prove that [XY Z] ≥ 9[ABC] (where the square brackets denote the area of a triangle). 350. P QR touches AB, AC of 4ABC at Q, R respectively and ABC internally at P . Prove that the midpoint of QR is the incenter of 4ABC. 351. Let BD and CE be the bisectors of angles B and C in triangle ABC. The line DE meets the circumcircle of ABC in K and L, and I is the incenter of ABC. Show that RIKL = 2RABC , where RXY Z is circumradius of triangle XY Z. 352. Let ABC be a triangle, H its orthocenter, O its circumcenter, and R its circumradius. Let D be the reflection of A across BC, E that of B across CA, and F that of C across AB. Prove that D, E and F are collinear if and only if OH = 2R. 353. Let ABC be a triangle and let P be a point in its interior. Lines P A, P B, P C intersect sides BC, CA, AB at D, E, F , respectively. Prove that [P AF ] + [P BD] + [P CE] =
1 [ABC] 2
354. In 4ABC, BD and CE are the angle bisectors of ∠B and ∠C, cutting CA, AB at D, E respectively. If ∠BDE = 24◦ and ∠CED = 18◦ , find the angles of ABC.
33
355. Let ABC be a triangle, D, E and F the points of tangency of the incircle with sides BC, CA, AB respectively. Let P be the second point of intersection of AD and the incircle. If BF P C is a cyclic quadrilateral prove that P E is parellel to BC. 356. A triangle P QR is inscribed in a triangle ABC with P on BC, Q on CA, R on AB. Start with a point P1 on BC distinct from P . Define a sequence of points Pi , Qi , Ri , i = 1, 2, 3, . . . on BC, CA, AB respectively such that Pi Qi , Qi Ri and Ri Pi+1 are parallel to P Q, QR, RP respectively. Prove that the sequence will close after finite steps (i.e. some Pi will coincide with an earlier Pj ) iff AP , BQ, CR are concurrent. 357. Let H be the orthocenter of acute triangle ABC with circumradius and inradius R, r respectively. Prove that AH + BH + CH = 2R + 2r 358. In a triangle ABC, O is the circumcenter. Midpoints of OA, OB, OC are X, Y , Z. A0 , B 0 , C 0 are mid-points of BC, CA, AB, then prove that the circles A0 OX, B 0 OY , C 0 OZ are concurrent. And the concurrency point lies on the line joining the circumcenter and Kosnita point of ABC. 359. Let Ia , Ib , Ic be the excentres of ABC corresponding to vertices A, B, C. H, I, O, Na are the orthocentre, incentre, circumcentre and Nagel point of ABC. Let S be the point such that O is the midpoint of HS. Prove that the centroid of triangles Ia Ib Ic and SINa coincide. 360. In a triangle 4ABC, AX is a symmedian with X lying on the circumcircle of 4ABC. M be the midpoint of AX. Extended BM and CM meet AC and AB at K and L respectively. Prove that the circumcenter of 4AKL lies on the line joining A and the nine-point center of 4ABC. 361. let S = sinn (a) + sinn (60◦ − a) + sinn (60◦ + a), ∀n ∈ N. Find all such n numbers for whom S does’t depent on a. 362. Circle A has radius 3 and circle B has radius 2. Their centers are 13 units from one another. A common external tangent P Q is drawn such that P lies on circle A and Q lies on circle B. Circles A0 and B 0 are constructed outside circles A and B such that circle A0 is tangent to AB, P Q, and circle A, and circle B 0 is tangent to AB, P Q, and circle B. What is the distance between the centers of A0 and B 0 ? 363. A circle ω touches the sides AB, AC of isosceles tringle ABC and interseects the side BC at K, L. AK intersects the circle ω secondly at point M . The points P , Q are symetric to point K depends on B, C respectively. Prove that the circumcircle of P M Q touches the circle ω.
34
364. Let ABCDEF G be a convex heptagon. Let BD intersects GE at A1 , and similarly define B1 , C1 , D1 , E1 , F1 , G1 . Prove that AB1 BC1 CD1 DE1 EF1 F G1 GA1 · · · · · · =1 A1 B B1 C C1 D D1 E E1 F F1 G G1 A 365. In a cyclic quadrilateral ABCD, AC ∩ BD = E and AB ∩ CD = F . EX is the E-symmedian of 4EAD and EY is the E-symmedian of 4EBC with X, Y lying on AD, BC respectively. Prove that XY k EF . 366. In a convex quadrilateral ABCD the points P and Q are chosen on the sides BC and CD respectively so that ∠BAP = ∠DAQ. Prove that the line, passing through the orthocenters of triangles ABP and ADQ is perpendicular to AC if and only if the triangles ABP and ADQ have the same areas. 367. Consider a point P outside a circle ω. Tangents P X, P Y are drawn from P to ω. Take a line passing through P intersecting ω at A and B. Through Y draw a line parallel to AB to intersect ω again at C. Prove that CX bisects AB. 368. An acute triangle ABC has altitudes BB 0 , CC 0 . Let L, M , N be the midpoints of B 0 C 0 , C 0 B, B 0 C respectively. Let K be the intersection of the line through M perpendicular to LB and the line through N perpendicular to LC. Prove that KB = KC. 369. Consider a circle ω and a point P outside it. Tangents P X, P Y are drawn from P to ω. Consider a point A on the smaller arc XY . AX, AY are joined to meet P Y , P X at B, C respectively. Two circles ω1 , ω2 are drawn centered at B, C and with radius BY , CX respectively. Suppose, one of the intersection points of ω1 , ω2 is D. Prove that ∠BDC = 60◦ . 370. ABCD is a convex quadrilateral with AB not parallel to CD. A circle through A and B touches at X and a circle through C and D touches AB at Y . The two circles meet at U and V . Show that U V bisects XY iff BC is parallel to DA. 371. A1 , A2 are points on BC, B1 , B2 on AC, C1 , C2 on AB, such that AA1 , BB1 , CC1 are concurrent and AA2 , BB2 , CC2 are concurrent. We know that AB ∩ A1 B1 , BC ∩ B1 C1 , CA ∩ C1 A1 are collinear and call that line as `1 . Similarly for A2 , B2 , C2 define `2 . (1) Prove that A, `1 ∩ `2 , B1 C1 ∩ B2 C2 are collinear (2) BC ∩ B2 C2 , CA ∩ C2 A2 , AB ∩ A1 B1 are collinear. 372. Let ABC be an acute triangle. Suppose a circle ω1 , with center O1 , touches the sides BC produced at E, AC produced at G, and AB at C 0 . Suppose also that another circle ω2 , with center O2 , touches the sides AB produced at H, BC produced at F , and AC at B 0 . Let the extensions of EG and F H intersect at P . Prove that: P A ⊥ BC. 35
373. In a triangle ABC, squares ABDE and ACF G are drawn outside the triangle. Perpendiculars BX, CY are drawn on BC to meet EG at X, Y . Prove that BX + CY ≥ BC + EG. 374. Let ABCD be a cyclic quadrilateral. Let AB intersects DC at E and BC intersects AD at F . Extend AC and DB to meet EF at X and Y respectively. Prove that if XB = XD, then Y A = Y C. 375. In a cyclic quadrilateral ABCD, M is the mid-point of AD. Take a BN point N on M BC such that BM M C = N C . Suppose, AB ∩ CD = E and AC ∩ BD = F . Prove that E, F , N are collinear. 376. Let ABCD be a quadrilateral such that AB + CD = BC + DA. Let ω1 be a circle tangent to the sides AB and BC, and ω2 is a circle tangent to the sides CD and DA. Prove that if there is a line through A tangent to ω1 and ω2 , then there is a line through C tangent to ω1 and ω2 as well. 377. In a triangle ABC, A1 B1 C1 is the in-touch triangle. Tangents Ta , Tb , Tc are drawn at La , Lb , Lc on the incircle where La , Lb , Lc lie on the smaller arcs B1 C1 , C1 A1 , A1 B1 respectively. Suppose, Ta intersects BC at Ka and B1 C1 at Ka0 . Analogously define Kb , Kb0 , Kc , Kc0 . Prove that Ka , Kb , Kc are collinear iff Ka0 , Kb0 , Kc0 are collinear. 378. I is the incenter of the triangle ABC. Prove that the circumcenter of the triangle AIB lies on CI. 379. A circle K is tangent to two parallel lines `1 and `2 . A second circle K1 is tangent to `1 at A and to K externally at C. A third circle K2 is tangent to `2 at B and to K externally at D and to K1 at E. Let Q be the intersection of AD and BC. Prove that QC = QD = QE. 380. In a triangle ABC, take points D, E on BC such that ∠BAD = ∠ACB, ∠CAE = ∠ABC. Suppose, M , N are midpoints of AB, AC. Prove that M D and N E meet on BOC, where O is the circumcenter of ABC. 381. Two sides of a triangle are 6 and 7 units. The sum of the cirumradius and the third side is 13 units. What is the geometric mean of the minimum and the maximum areas of the triangle? Note: both the third side of the triangle and the circumradius have integer lengths. 382. Let ABC be an acute angled triangle. Let AD be the altitude on BC, and let H be any interior point on AD. Lines BH and CH, when extended, intersect AC and AB at E and F , respectively. Prove that ∠EDH = ∠F DH. 383. In a triangle ABC a parallel line to the side BC cuts the side AB and AC at D and E respectively. Let P be any point inside the triangle ADE. BP and CP cuts the line DE at F and G. If the circles P DG and P EF meet at Q other than P then show that A, P , Q are collinear.
36
384. Right triangle ABC has one angle of a measure of 45 degrees. D is the midpoint of AB. DB is the hypotenuse of 4BDE, which has an angle with a measure of 60 degrees, which is angle DBE. Prove that √ 2 6 DE × AC = AB 2 × 3 385. Let I be the incenter of triangle ABC. Prove that (IA)(IB)(IC) = 4Rr2 , where R is the circumradius of ABC and r is the inradius of ABC. 386. Choose point P on side AB of 4ABC. Let the line parallel to BC through P meet AC in Q, the line parallel to AB through Q meet BC in R, the line parallel to CA through R meet AB in S, the line parallel to BC through S meet AC in T , and the line parallel to AB through T meet BC in U . Prove that P U k AC. 387. Let 4ABC be a triangle and P is a varying point on the arc BC of ABC. Show that circle through P and the incenters of 4P AB and 4P AC passes through a fixed point independent of P . 388. In 4ABC, AB 6= AC. Point N is on CA, point M is on AB, point P and Q are on BC such that M P is parallel to AC and N Q is parallel to AB. CQ Given that BP AB = AC and A, M , Q, P , N are concyclic, find ∠BAC. 389. A cube is inscribed in a regular tetrahedron of unit length in such a way, that each of the 8 vertices of the cube lie on faces of the tetrahedron. Determine the possible side lengths for such a cube. 390. Let A1 , B1 , C1 denote the feet of the perpendiculars dropped from the vertices A, B, C of a triangle ABC on the line `. Prove that the perpendiculars dropped from A1 , B1 , C1 on BC, CA, AB respectively, are concurrent. 391. Consider a triangle ABC. For each circle K passing through A and B, define Pk , Qk to be the intersections of K with BC and AC respectively. (a) Show that for any circles K, K 0 passing through A and B, the lines Pk Qk and Pk0 Qk0 are parallel. (b) Determine the locus of the circumcircle of 4CPk Qk . 392. ABCD is an isosceles trapezium with AB k CD. The inscribed circle ω of 4BCD meets CD at E. Let AF be the internal angle bisector of ∠DAC such that EF ⊥ CD. Let ACF meet line CD at C and G. Prove that 4AF G is isosceles. 393. Given n ≥ 3 points on a plane, where not all of them lie on a line, prove that we can choose 3 non collinear points such that its circumcircle contains all other points.
37
394. A triangle ABC in the plane π is said to be good if it has the following property: for any point D in space, out of the plane π, it is possible to construct a triangle with sides of lengths |AD|, |BD| and |CD|. Find all good triangles. 395. Let ABCDE be a convex pentagon, and let F = BC ∩DE, G = CD∩EA, H = DE ∩ AB, I = EA ∩ BC, J = AB ∩ DE. Suppose that the areas of the triangles AHI, BIJ, CJF , DF G, EGH are all equal. Then the lines AF , BG, CH, DI, EJ are all concurrent. 396. Six points A1 , A2 , A3 , A4 , A5 , A6 lie on a line ` in that order. Two circular arcs γi , βi , connect the points Ai and Ai+3 for each 1 ≤ i ≤ 3. These arcs lie on the same side of `. Call the curved quadrilateral formed by the circles γi , βi , γj , βj as Ci,j . There are three such quadrilaterals. Prove that if any two of them is circumscribed, then so is the third one. 397. Let γ and ω be 2 internally tangent circles at T , where ω is inside γ. C is a point inside γ but outside ω. The 2 rays from C tangent to ω intersects γ at A, B, and tangent to ω at X, Y . Let M be the midpoint of the arc AB containing T . Prove that XY , AB, T M are concurrent. 398. Let ABCD is a square inscribed in a circle of radius 1, so that there is a vertex in each radius and two vertices that surround the arc tangent. If the central angle is 2α, determine the value of α closest that makes maximum square area. 399. Let ABCD be a convex quadrilateral, P = AB ∩ CD, Q = AD ∩ BC, O = AC ∩ BD. Show that, if ∠P OQ is a right angle, then P O is the angle bisector of ∠AOD and QO is the angle bisector of ∠AOB. 400. Let P be the Fermat point of triangle ABC. Show that the Euler lines of triangles AP B, AP C and BP C concur at a single point. 401. Construct triangle ABC using straightedge and compass, given b − c, hc − hb and r. (b, c are sides, hb , hc are altitudes, r is the inradius). 402. Let ABC be triangle. ABB1 A2 , BCC1 B2 , CAA1 C2 squares are consturcted outside ABC. Prove that perpendicular bisectors of A1 A2 , B1 B2 , C1 C2 are concurrent. 403. Consider 4ABC, such that AC > AB. Let Ma , Mb , Mc be its respective Malfatti circles; let Ma and Mc touch at E; let Mb touch AB at D; let S1 be the circle passing through D and touching Ma and Mc at E; let K be the centre of S1 ; let M be the centre of Mc ; let CM and AK intersect at L; let N be the orthogonal projection of L on AB. Prove that (L, LN ) touches the Nine-Point circle of 4ABC. 404. In a triangle ABC squares are drawn outwardly on the sides BC, CA, AB with their centres Oa , Ob and Oc respectively. Show that AOa , BOb and COc are concurrent. 38
405. Let ABCDE be pentagon such that BC k AD , BD k AE. Let M and N be midpoints of CD and DE sides ,respectively. If O is intersection of BN and AM lines, prove [M DN O] = [ABO]. 406. Let ABC be triangle. D and E points lies on on BC side such that ∠BAD = ∠CAE. Incircles of 4ABD, 4ACE tangents BC line at M and N . Prove that 1 1 1 1 + = + MB MD NC NE 407. Through vertex A of a tetrahedron ABCD passes a plane tangent to the circumscribed sphere of the tetrahedron. Show that the lines of intersection of the plane with the planes ABC, ABD, ACD, form six equal angles if and only if: AB · CD = AC · BD = AD · BC. 408. Sasha has a compass with fixed radius s and Rebecca has a compass with fixed radius r. Sasha draws a circle (with his compass) and Rebecca then draws a circle (with her compass) that intersects Sasha’s circle twice. We call these intersection points C and D. Charlie draws a common tangent to both circles, meeting Sasha’s circle at point A and Rebecca’s circle at point B, and then draws the circle passing through A, B, and C. Prove that the radius of Charlie’s circle does not depend on where Sasha and Rebecca choose to draw their circles, or which of the two common tangents Charlie draws. 409. Let A, B, C, D be points in space, let M be the midpoint of AC, and let N be the midpoint of BD. Prove that 4M N 2 = AB 2 + BC 2 + CD2 + DA2 − AC 2 − BD2 410. Let A1 A2 . . . An be a polygon, not necessarily convex, with perimeter 4, i.e. A1 A2 + A2 A3 + · · · + An A1 = 4. Prove that there exists a point P such that P Ai ≤ 1 for all 1 ≤ i ≤ n. 411. Let E be a point inside the triangle ABC such that ∠ABE = ∠ACE. Let F and G be the feet of perpendiculars from E to the internal and external bisectors respectively, of ∠BAC. Prove that the line F G passes through the mid-pint of BC. 412. Let 4ABC be an acute triangle with ∠BAC = 30◦ . The internal and external angle bisectors of ∠ABC meet the line AC at B1 and B2 , respectively, and the internal and external angle bisectors of ∠ACB meet the line AB at C1 and C2 , respectively. Suppose that the circles with diameters B1 B2 and C1 C2 meet inside the triangle ABC at point P . Prove that ∠BP C = 90◦ . 413. Let ABC be an acute triangle with circumcircle Γ. Let P be a fixed point on Γ. ` is a variable line through P . Let `a , `b , `c be the reflections of 39
` about BC, CA, AB respectively. Prove that the circumcircle of the triangle determined by the lines `a , `b , `c passes through a fixed point independent of `. 414. Given any four lines show that there exists a unique point such that feet of the perpendiculars drawn from this point to these lines are collinear. 415. Let ABCD be a cyclic quadrilateral with circumcircle Γ. Suppose w1 , w2 are 2 distinct circles internally tangent to Γ and tangent to AB, CD. Let the midpoints of minor arcs AB, CD be M , N . Suppose w1 , w2 have centers O1 , O2 . Prove that M N ⊥ O1 O2 . 416. An isosceles right-angled triangle T is given. Consider a variable circle C. Prove that, T ∪ C − T ∩ C will be minimum when center of C will lie on the main altitude of T and it will divide the altitude in golden ratio. 417. In a triangle ABC, I is the incenter. AI, BI, CI intersects the opposite sides at A0 , B 0 , C 0 . Prove that if A0 B 0 C 0 is equilateral then ABC is equilateral. 418. Let ABCD be a cyclic quadrilateral with circumcircle Γ. Suppose w1 , w2 are 2 distinct circles internally tangent to Γ and tangent to both AB, CD. Let the midpoints of minor arcs AB, CD be M , N . Prove that M N is the radical axis of w1 , w2 . 419. ABC is a triangle. X, Z are on rays BC, BA such that BA = BX and BC = BZ. Let the excircle of 4ABC opposite B touch the extension of BC at S and the side AC at T . If XT intersects CZ at Y , prove that AY S is a straight line. 420. If the Kosnita point and De-Longchamps point of a triangle coincides, then the triangle is equilateral. 421. Let I be the incentre of a triangle ABC and Γa be the excircle opposite A touching BC at D. If ID meets Γa again at S, prove that DS bisects ∠BSC. 422. ABC is a right triangle at A. D is a point such that DC ⊥ BC. Let M be the midpoint of BC, and DM intersects BA at E. Point F is on DA such that BF k CE. Prove that ∠ECF = 90◦ . 423. Suppose, the circle touching AB, AC and ABC internally touches ABC at A1 . I is the incenter of ABC. Suppose, IA1 ∩ BC = A2 . Similarly define B2 , C2 . (a) Prove that, AA2 , BB2 , CC2 are concurrent. (b) Suppose, A02 is the harmonic conjugate of A2 wrt B, C. Similarly define B20 , C20 . Then prove that, A02 , B20 , C20 lie on the polar of the orthocenter of A0 B 0 C 0 wrt the incircle of ABC, where A0 B 0 C 0 is the in-touch triangle of ABC. 40
424. Let S1 and S2 be two circles that intersect at M and N , with centers O1 and O2 respectively. The line t is the common tangent to S1 and S2 closer to M . The points A and B are the intersection points of t with S1 and S2 , C is the point such that BC is a diameter of S2 , and D the intersection point of the line O1 O2 with the perpendicular line to AM through B. Show that M , D and C are collinear. 425. In a triangle ABC the A-excircle touches the side BC at A0 . Analogously define B 0 , C 0 . Prove that the lines through A0 , B 0 , C 0 parallel to the corresponding angle bisector are concurrent. 426. ABC is an isosceles triangle at A. A circle Γ is tangent to AC at E, tangent to AB at F , and intersects the side BC at a point D. Let AD intersect Γ again X. Let F X intersect DE at Y , EX intersect DF at X. Prove that BY , CZ, EF are concurrent. 427. In a triangle ABC, A1 , B1 , C1 are the diametrically opposite points of A, B, C wrt ABC. A2 , B2 , C2 are reflections of A1 , B1 , C1 on BC, CA, AB. Prove that AA2 , BB2 , CC2 are concurrent. 428. P , Q are two arbitrary points in the plane of the equilateral 4ABC. P1 , P2 , P3 are the orthogonal projections of P on BC, CA, AB and Q1 , Q2 , Q3 are the orthogonal projections of Q on BC, CA, AB. Show that one of the segments P1 Q1 , P2 Q2 , P3 Q3 equals the sum of the other two. 429. ABC is an equilateral triangle and ` is a line tangent to its circumcircle. Let the distances √ √from√ A, B, C onto ` be d1 , d2 , d3 . Prove that one of the values d1 , d2 , d3 equals the sum of the other two. 430. Let P1 · · · Pn be a planar convex polygon such that for any i 6= j there exists some k such that ∠Pi Pk Pj = 60◦ . Prove that our polygon is just an equilateral triangle. 431. In a triangle ABC, I is the incenter. Take a point A1 on BC such that ∠AIA1 = 90◦ . Suppose, A2 is the isotomic point of A1 wrt BC. Analogously define B2 , C2 . (a) Prove that A2 , B2 , C2 are collinear. (b) Suppose, A2 , B2 , C2 lie on `. A0 B 0 C 0 be the intouch triangle of ABC. Prove that IP ⊥ ` where P is the Prasolov point of the medial triangle of A0 B 0 C 0 . 432. Given are three circles (O1 ), (O2 ), (O3 ), pairwise intersecting each other, that is, every single circle meets the other two circles at two distinct points. Let (X1 ) be the circle externally tangent to (O1 ) and internally tangent to the circles (O2 ), (O3 ); circles (X2 ), (X3 ) are defined in the same manner. Let (Y1 ) be the circle internally tangent to (O1 ) and externally tangent to the circles (O2 ), (O3 ), the circles (Y2 ), (Y3 ) are defined in the same way. Let (Z1 ), (Z2 ) be two circles internally tangent to all three circles 41
(O1 ), (O2 ), (O3 ). Prove that the four lines X1 Y1 , X2 Y2 , X3 Y3 , Z1 Z2 are concurrent. 433. Given n ≥ 2 points in a plane, determine the maximum number of lines of symmetry. 434. Let ABC be an acute triangle with its altitudes BE, CF . M is the midpoint of BC. N is the intersection of AM and EF . X is the projection of N on BC. Y , Z are respectively the projections of X onto AB, AC. Prove that N is the orthocenter of triangle AY Z. 435. The point A is on a circle with center O. The line OA is extended to C so that OA = AC, and B is the midpoint of AC. The point Q is on the circle O such that ∠AOQ is obtuse. The line QO meets the perpendicular bisector of CQ at P . Show that ∠P OB = 2∠P BO. 436. In a triangle ABC, G is the centroid. AG, BG, CG meet ABC at A1 , B1 , C1 . Tangents are drawn at A1 , B1 , C1 on ABC to form a triangle A2 B2 C2 . Suppose, AG intersects B1 C1 at A3 . Define, B3 , C3 analogously. Prove that A2 A3 , B2 B3 , C2 C3 are concurrent at the Euler line of ABC. 437. Incircle (I) of 4ABC touches BC, CA, AB at D, E, F , respectively. Internal bisector of ∠BIC cuts BC at M and AM cuts EF at P . Show that DP bisects ∠EDF . 438. Prove that, in a triangle ABC isogonal point of Georgenne point lies on OI where O and I are the circumcenter and incenter of ABC. 439. In a triangle ABC, (I) is the incircle. Let X, Y , Z be the reflections of I wrt BC, CA, AB. X 0 , Y 0 , Z 0 are the midpoints of the smaller arcs BC, CA, AB. Let XX 0 , Y Y 0 , ZZ 0 meet the circumcircle of 4ABC again at X 00 , Y 00 , Z 00 . Show that I is the incenter of 4X 00 Y 00 Z 00 . 440. In a triangle ABC, BB1 and CC1 are internal angle-bisectors. Suppose, one of the intersection points of B1 C1 and ABC is K. Ia , Ib , Ic are the excenters of ABC. Suppose, BC intersects Ia Ib Ic at some point L. Prove that Ia L = Ia K. 441. Three congruent circles pairwise intersect at A, A0 ; B, B 0 and C, C 0 respectively. Prove that the perpendiculars from A0 , B 0 , C 0 to BC, CA, AB are concurrent. 442. Anti-cevian triangle of the circumcenter of a triangle is homothetic to the cevian triangle of the isotomic conjugate of its orthocenter. 443. Let AB be a diameter and BC be a chord of a circle ABC. Bisect the minor arc BC at M ; and draw a chord BN equal to half of the chord BC. Join AM . Describe two circles with A and B as centers and AM and BN as radii, cutting each other at S and S 0 , and cutting the given circle again at the points M 0 and N 0 respectively. Join AN and BM intersecting at R, 42
and also join AN 0 and BM 0 intersecting at R0 . Through B draw a tangent to the given circle, meeting AM and AM 0 produced at Q and Q0 respectively. Produce AN and M 0 B to meet at P , and also produce AN 0 and M B to meet at P 0 . Show that the eight points P, Q, R, S, S 0 , R0 , Q0 , P 0 are cyclic, and that the circle passing through these eight points is orthogonal to the given circle ABC. 444. Let ABC be a scalene triangle. A circle (O) passes through B, C, intersecting the line segments BA, CA at F , E respectively. The circumcircle of triangle ABE meets the line CF at two points M , N such that M is between C and F . The circumcircle of triangle ACF meets the line BE at two points P , Q such that P is betweeen B and E. The line through N perpendicular to AN meets BE at R; the line through Q perpendicular to AQ meets CF at S. Let U be the intersection of SP and N R, V be the intersection of RM and QS. Prove that three lines N Q, U V , RS are concurrent. 445. In a triangle ABC, H is the orthocenter and (O) is its circumcircle. Prove that isotomic line of the polar of H wrt (O) is perpendicular to HK where K is the Kosnita point of ABC. 446. In a triangle ABC, A0 B 0 C 0 is its reflection triangle. Take A1 on B 0 C 0 such B 0 A1 sin 2C 0 0 that C 0 A1 = sin 2B . Analogously define B1 , C1 . Prove that A A1 , B B1 , 0 C C1 are concurrent on the Euler line of ABC. 447. ABCD is a convex cyclic quadrilateral. E ≡ AD ∩BC and F ≡ AC ∩BD. Show that the orthocenters of 4ECD, 4EAB, 4F CD and 4F AB are vertices of a parallelogram. 448. Suppose the 2 convex circumscribed quadrilaterals ABCD and AECF have the same incenter, with E, F inside ABCD. Prove that BE + ED = BF + F D. 449. Prove that Schiffler point of a triangle lies on its Euler line. 450. Points M , N , K, L are taken on the sides AB, BC, CD, DA of a rhombus ABCD, respectively, in such a way that M N k LK and the distance between M N and KL is equal to the height of ABCD. Show that the circumcircles of the triangles ALM and N CK intersect each other, while those of LDK and M BN do not 451. As shown in the figure, M , N , T are tangent points of the corresponding segments and circles, I is the incenter of 4ABC, prove that I lies on the circumcircle of T BM and T CN .
43
452. In a triangle ABC, I, I 0 , N , F are its incenter, Speiker center, nine-point center and Feurbach point. If A0 B 0 C 0 is the medial triangle of ABC and A1 B1 C1 be the circumcevian triangle of I 0 wrt A0 B 0 C 0 , then prove that, isogonal conjugate of the point at infinity on N I 0 wrt A1 B1 C1 is F . 453. D is an arbitrary point on the sideline BC of 4ABC. ` is the parallel to BC through A. P is an arbitrary point on AD. BP cuts `, AC at B1 , B2 , respectively and CP cuts `, AB at C1 , C2 , respectively. Lines C1 B2 and B1 C2 meet at Q and AQ cuts BC at E. Show that BD = CE. 454. In a triangle ABC, a point P is taken such that if P A ∩ BC = A0 , P B ∩ CA = B 0 , P C ∩ AB = C 0 , then P A0 = P B 0 = P C 0 . If the perpendiculars from A, B, C to B 0 C 0 , A0 C 0 , A0 B 0 meet at Q, prove that Q is the Nagel point of the tangential triangle of A0 B 0 C 0 . 455. Incircle (I) of 4ABC is tangent to BC, CA, AB at D, E, F . The Aexcircle is tangent to BC at P . Let Q be the antipode of D wrt (I) and define the points M ≡ QE ∩ BC, N ≡ QF ∩ BC and R ≡ IA ∩ EF . Prove that the lines QR and P I meet at the symmedian point of 4QM N . 456. In an equilateral triangle ABC, P is a point inside ABC, AP , BP , CP intersect BC, CA, AB at A0 , B 0 , C 0 . And foot of the perpendiculars from P to BC, CA, AB are A1 , B1 , C1 . Prove that A0 B 0 C 0 is similar to A1 B1 C1 (directly or indirectly) iff P is center of ABC. 457. The incircle (I) of 4ABC touches BC and AC at A1 and B1 respectively. Points A2 and B2 are antipodes of A1 and B1 with respect to (I). Let
44
A3 and B3 be the intersection points of AA2 with BC and BB2 with AC respectively. Let M be the midpoint of AC ad N be the midpoint of A1 A3 . Line M I meets BB1 at T and line AT meets BC in P . Let Q ∈ BC, R be the intersection of AB and QB1 and N R ∩ AC = S. Prove that AS = 2SM if and only if BP = P Q. 458. The incircle of triangle ∆ABC touches sides BC, CA, AB at D, E, F respectively. Let P be any point inside triangle ∆ABC, and let X, Y , Z be the points where the segments P A, P B, P C, respectively, meet the incircle. Prove that the lines DX, EY , F Z are concurrent. 459. There are given n + 1 points P1 , P2 , · · · , Pn and Q in a plane, n ≥ 3, no 3 of which are collinear. It is known that for each pair of distinct points Pi and Pj one can find a point Pk such that Q lies in the interior of 4Pi Pj Pk . Show that n is odd. 460. In parallelogram ABCD, ∠BAC = 40◦ and ∠BCA = 20◦ . Let G and H be points on AC and AD, respectively, such that ∠ABG = ∠AHG = 90◦ . Let E and F be points on AC and AD, respectively, such that B, E, F are collinear and AF = EG. Prove that AF = HD. 461. Suppose a circle passes through the feet of the symmedians of a nonisosceles triangle ABC, and is tangent to one of the sides. Show that a2 + b2 , b2 + c2 , c2 + a2 are in geometric progression when taken in some order. 462. Circle O is the circumcircle of triangle ABC, M , F lie on segment AB and N , E on AC, such that, AM = AN , P F = P M , QN = QE, P , Q are the intersection of M N and circle (O), G is the midpoint of P Q, BG meets P F at R and CG meets QE at S. Prove that, RS k BC. 463. Circle (O1 ), (O2 ) (R1 > R2 ) meets at P , Q, O1 T is tangent to Circle (O2 ) at T , QT , P T , QO2 intersect circle (O1 ) at R, S, M , respectively. Let N , L be the intersections of O2 T and SM , circle (O2 ), respectively, T K is parallelled with RM and meets P L at K. Prove that RK passes the midpoint of M N . 464. I inscribes O at A, point B, C lies on O, AB, AC meet I at D, E. Let DF , EG be the tangencies to I that meet O at F , G above BC. BG meets CF at H, AK the bisector of ∠BAC meets BC at K, J is the midpoint of BC. Let O1 , O2 be the circumcenters of (AOJ) and (AHK),respectively. Prove that O and O2 are symmetric about O1 . 465. Let Γ(I, r) and Γ(O, R) be the incircle and circumcircle, respectively, of triangle ABC. Consider all triangles Ai Bi Ci which are simultaneously inscribed in Γ(O, R) and circumscribed to Γ(I, r). Prove that the centroids of the triangles Ai Bi Ci lie on a circle.
45
466. Given a triangle ABC. Let X, Y , Z be points on the segments BC, CA, \ = CBY \ = ACZ. [ AB, respectively, such that AZ = BX = CY and BAX Prove that triangle ABC is equilateral. 467. Given a 4ABC with D, E, F lie on BC, AC, AB, respectively, such that AD, BE, CF are concurrent at I and DE ⊥ DF . The circumcircle of 4ABI and 4ACI meet BC at G, H, J lies on segments AB and K on AC such that J, I, K are collinear. Prove that HI GI + =1 AJ AK 468. Let P and Q be such points on sides AB and AC of a triangle ABC respectively, that AP = BQ. Prove that if AB is the longest side of the triangle, then 2 · P Q > BC. 469. In radian measure, prove that n X cos2 α2i 1 2 A +A× > n2 2 αi i=1
where A is the sum of angles of a convex polygon A1 A2 . . . An and αi are its interior angles. (n ≥ 3, n ∈ N ). 470. In triangle ABC of unit area. P , Q, R are on sides BC, CA, AB. Prove that all three areas of triangles AQR, BP R, CP Q are not greater than 14 . 471. Trilinear polar of a point P wrt a triangle ABC is perpendicular to P 0 G where P 0 is the isogonal conjugate of P wrt ABC and G is the centroid of the pedal triangle of P 0 wrt ABC. 472. Let ω1 , ω2 , ω3 , ω4 be four circles in the plane. Suppose that ω1 and ω2 intersect at P1 and Q1 , ω2 and ω3 intersect at P2 and Q2 , ω3 and ω4 intersect at P3 and Q3 , and ω4 and ω1 intersect at P4 and Q4 . Show that if P1 , P2 , P3 , P4 lie on a line or circle, then Q1 , Q2 , Q3 , Q4 also lie on a line or circle. 473. Let ABCD be a parallelogram with side lengths AB = a, AD = 1, and with ∠BAD = α. If 4ABD is acute, prove that the four circles of radius 1 with centers A, B, C, D cover the paralleogram if and only if √ a ≤ cos α + 3 sin α 474. ω1 and ω2 with unequal radiis and centers O1 , O2 meets at A. Tangency of ω2 in A meets ω1 at B, tangency of ω1 in A meets ω2 at C. BC meets ω1 and ω2 respectively at D, E differ from B, C. Let P be the intersection of AD and ω2 and Q be that of AE and ω1 . Prove that,quadrilateral O1 O2 O3 O4 is an isosceles trapezoid, where O3 , O4 are the circumcenters of 4P BD and 4QEC, respectively. 46
475. In a trapezoid ABCD with AD k BC, O1 , O2 , O3 , O4 denote the circles with diameters AB, BC, CD, DA, respectively. Show that there exists a circle with center inside the trapezoid which is tangent to all the four circles O1 , O2 , O3 , O4 if and only if ABCD is a parallelogram. 476. Let ABCD be a convex quadrilateral. Suppose that the external bisectors of ∠DAC and ∠DBC meet at P . Show that AD + AC = BD + BC if and only if ∠DP A = ∠CP B. 477. D is the midpoint of BC of 4ABC with ω its circumcircle. AD meets ω at E and DF which is parallel to AB meets CE at F . BF meets ω again at G and H is the intersection of AB and DG. Prove that CH = AC. 478. R is the midpoint of BC of 4ABC with ω its circumcircle, I the incenter and Q the B-excenter. AQ meets ω at P and P C meets QR at S. Prove that IS is the bisector of ∠P SR ⇐⇒ P Q = QR. 479. Suppose that A0 B 0 C 0 is the orthic triangle of a triangle ABC and let P be a point inside ABC. Suppose A1 , B1 , C1 are the orthogonal projections of A, B, C onto OP , where O is the circumcentre of ABC. Let `1 , `2 , `3 be the reflections of the lines A0 A1 , B 0 B1 , C 0 C1 in the altitudes AA0 , BB 0 , CC 0 , respectively. Let k1 , k2 , k3 be the lines through A, B, C and parallel to `1 , `2 , `3 respectively. Then prove that the lines ki concur at a point on the nine point circle of triangle ABC. 480. Given triangle ABC with circumcircle ω, the circle ωa touches AB and AC at D1 and D2 and touches ω internally at L. Define E1 , E2 , M , and F1 , F2 , N in a corresponding way. Prove that (a) AL, BM , CN are concurrent. (b) D1 D2 , E1 E2 , F1 F2 are concurrent and the point of concurrency in the incentre of 4ABC. 481. Let O be an interior point of 4ABC such that AB + BO = AC + CO. Suppose that P is a variable point on the side BC, and that Q and R are points on AB and AC, respectively, such that P Q k CO and P R k BO. Prove that the perimeter of quadrilateral AQP R is constant. 482. Show that for every convex n-gon (n ≥ 4), the arithmetic mean of the lengths of its sides is less than the arithmetic mean of the lengths of all its diagonals. 483. Let R, r be respectively the circumradius and inradius of a regular 1986gonal pyramid. Prove that R 1 ≥1+ π r cos 1986 and find the total area of the surface of the pyramid when the equality occurs. 47
484. let ABC be a triangle. Point D, E, F are on sides BC, CA, AB, respectively, such that DC + CE = EA + AF = F B + BD. Prove that DE + EF + F D ≥
1 (AB + BC + CA) 2
485. ω is a semicircle where I the center lies on side BC of 4ABC (AB > AC) where ∠BAC = 90◦ and AB, AC are its tangences. H is on ω such that AH ⊥ BC. Let P be the projection of B on the parallel of HI through C. Bisector of ∠BP C meets the parallel of BC through A at Q. R is the pedal of B on P Q and P Q meets BC at S. Prove that P S = 2 · QR. 486. Prove that it is possible to construct a hexagon in a convex polygon such that the area of the hexagon is at least 43 of the polygon. 487. In triangle ABC the bisector from A meets the circumcircle at point K. If X is the midpoint of AK prove that BX + CX ≥ AK. 488. Let ABCDEF be a convex hexagon such that BCEF is a parallelogram and ABF an equilateral triangle. Given that BC = 1, AD = 3, CD + DE = 2, compute the area of ABCDEF . 489. Let Γ be the circumcircle of a triangle ABC. Suppose its incircle touches the side BC at A1 , and let A2 be the midpoint of BC. Let wa be the circle centered at A2 and passes through A1 . Similarly define wb , wc . Prove that if wa is tangent to the arc BC of Γ not containing A, then either wb is tangent to the arc AC of Γ not containing B or wc is tangent to the arc AB of Γ not containing C. 490. Prove that if α, β are angles of a triangle acute-angled or rectangular, then: p √ p √ > cos α + cos β 2S where p =
a+b+c , 2
S is area.
491. Given is a convex quadrilateral ABCD. Let E be the intersection point of its diagonals. If 4ABE, 4BCE, 4CDE, 4DAE all have same inradius length, prove that ABCD is rhombus. 492. Points A, X, D lie on a line in this order, point B is on the plane such that \ > 120◦ , and point C is on the segment BX. Prove the inequality: ABX 2AD ≥
√
3(AB + BC + CD).
493. Let C1 be a point on line segment AB such that the inradii of 4C1 AC is equal to that of 4CC1 B. Define B1 and A1 similarly. Show that if CC1 , AA1 and BB1 are concurrent inside the 4ABC, then 4ABC is isosceles.
48
494. Let ω and I be the A-excircle and incenter of 4ABC where ω tangents BC, AB, AC at D, E, F ,respectively. P lies on AB and Q on AC such that P , Q, I are collinear and AP = AQ. AD meets EF at T and P T , QT meets BC at R, S,respectively. ω1 , ω2 are the circumcirles of 4T BS and 4T CR and ω1 , ω2 meets at U differs from T . Prove that U , I, T are collinear. 495. D is a point within the midian of BC of 4ABC. I the innercircle of 4BCD tangents BC, CD, BD at E, F , G, respectively. EG meets AB at M and EF meets AC at N , M F meets N G at H. Prove that HI ⊥ BC. 496. Let L be the perpendicular bisector of BC of 4ABC and bisector of ∠BAC meets L, BC at F , G, respectively. H lies on L and bisectors of ∠ABH, ∠ACH meets at P , BP , F P meet AH, CH at M , N ,respectively. Prove that M , G, N are collinear. 497. The circle ω and its chord BC are fixed and A lies on ω such that 4ABC is acute. CD ⊥ AB at E and DF k AC, where D lies on ω and F on AB. G lies on OF such that BG k OE. Find the track of G. \ = 45◦ . If E 498. In 4ABC, AD is the angle bisector (D ∈ BC) and ADB is the projection (orthogonal) of B onto AC and F the intersection of internal bisector of ∠AEB with AD, then 4ABF ∼ 4ACD. 499. (a) Prove that for any triangle s2 ≤
√ √ 1 (23 − 17) R2 + (4 + 17) r2 4
where s is the half of the perimeter. (b) Prove that ≤ becomes = for the equilateral triangle and for one more triangle. Find the angles of this triangle. 500. ω1 tangents AB, BC of 4ABC at D, F and ω2 tangents AC, BC at E, G where ω1 and ω2 are within 4ABC. AM tangents ω1 at M and AN tangents ω2 at N . F M meets GN at P and DF meets EG at Q. Prove that A, P , Q are collinear. 501. Show that a square can be inscribed in any regular polygon. 502. On the side AB of the convex pentagon ABCDE, consider a point F such that 4ADE ∼ 4ECF ∼ 4DBC. Prove that: AF EF 2 = BF CF 2 503. Let A0 , B 0 , C 0 be the midpoints of the sides BC, CA, AB, respectively, of an acute non-isosceles triangle ABC, and let D, E, F be the feet of the altitudes through the vertices A, B, C on these sides respectively. Consider 49
the arc DA0 of the nine point circle of triangle ABC lying outside the triangle. Let the point of trisection of this arc closer to A0 be A00 . Define analogously the points B 00 (on arc EB 0 ) and C 00 (on arc F C 0 ). Show that triangle A00 B 00 C 00 is equilateral! 504. A cyclic hexagon ABCDEF is such that AB ·CF = 2BC ·F A, CD ·EB = 2DE · BC and EF · AD = 2F A · DE. Prove that the lines AD, BE and CF are concurrent. 505. A triangular pyramid O(ABC) with base ABC has the property that the , OC+OA lengths of the altitudes from A, B and C are not less than OB+OC 2 2 OA+OB , respectively. Given that the area of ABC is S, calculate the and 2 volume of the pyramid. 506. We consider the points A, B, C, D, not in the same plane, such that AB ⊥ CD and AB 2 + CD2 = AD2 + BC 2 . (a) Prove that AC ⊥ BD. (b) Prove that if CD < BC < BD, then the angle between the planes (ABC) and (ADC) is greater than 60◦ . 507. Consider a triangle ABC and the natural number n ≥ 4. On the sides AB and AC are constructed in exterior regular poligons with n sides, having the centers O1 and O2 , respectively. On the side BC it is constructed in exterior an equilateral triangle with the center O3 . Prove that the triangle b = 120◦ . O1 O2 O3 is equilateral if and only if AB = AC and A 508. Let ABC be an acute triangle. Points D, E, and F lie on segments BC, CA, and AB, respectively, and each of the three segments AD, BE, and CF contains the circumcenter of ABC. Prove that if any two of the ratios BD CE AF BF AE CD DC , EA , F B , F A , EC , DB are integers, then triangle ABC is isosceles. 509. A square ABCD of side length 2 is given on a plane. The segment AB is moved continuously towards CD until A and C coincide with C and D, respectively. Let S be the area of the region formed by the segment AB while moving. Prove that AB can be moved in such a way that S < 5π 6 510. Let ABC be a triangle. Consider the points {M, N } ⊂ (BC) so that MB NB AB 2 and a point P ∈ (AM ). Prove that [ ∠ABP ≡ M C = N C + AC ∠CAM ⇐⇒ ∠CP M ≡ ∠CAN ] 511. In a triangle ABC, G and S are the centroid and symmedian point of ABC. P be a variable point. P 0 be the isotomic point of P wrt ABC and H be the orthocenter of ABC. Prove that HP 0 is perpendicular to the tri-linear polar of P wrt ABC iff P lies on GS. 512. For every point O on diameter AB of a given circle perform the following constructions. Let the perpendicular to AB at O meet the circle at P . Inscribe circles which are tangent to AB and OP and touches the given circle internally. 50
Let R and S be the points where the inscribed circles of the curvilinear triangle AOP and BOP touches AB respectively. Show that ∠RP S does not depends on the position of O. 513. In triangle ABC, ∠C > 10◦ and ∠B = ∠C + 10◦ . Points E, D are on line segments AB, AC respectively such that ∠ACE = 10◦ and∠ABD = 15◦ . Z is (different from A) intersection point of circumscribed circles of triangles ABD and AEC. Prove that ∠ZBA > ∠ZCA. 514. Triangles ABC and DBC share the common side BC. Let I, Q are inscribe circle’s center of 4ABC and 4DBC respectively. Prove that IQ ≤ AD. 515. For a circumscriptible quadrilateral ABCD with its incircle I, let AC cut BD at E. If three midpoints of AD, BC, EI are collinear, prove that AB = CD. 516. Let ABCD be a convex quadrilateral such that AB = AC = BD. Let the diagonals AC and BD intersect at O, and let the circumcircles of triangles ABC and ADO intersect at P , different from A. AP and BC intersect at Q. Prove that ∠COQ = ∠DOQ. 517. Given a triangle ABC, suppose, I is its incenter. B1 = BI ∩ AC and C1 = CI ∩ AB. Reflect B1 on the midpoint of AC to get B2 and reflect C1 on the midpoint of AB to get C2 . Prove that the Lemoine axis of 4BIC is anti-parallel to B2 C2 wrt ∠BIC. 518. In a triangle ABC for which 6(a + b + c)r2 = abc, we consider a point M on the inscribed circle and the projections D, E, F of M on the sides BC, CA and AB respectively. Let S, S1 denote the areas of the triangles ABC and DEF respectively. Find the maximum and minimum values of the quotient SS1 (here r denotes the inradius and, as usual, a = BC, b = CA, c = AB). 519. (1) A1 A2 A3 a non-isoscele triangle. (2) I ,it’s Incenter. (3) Ci (i = 1, 2, 3) a circle pass through I and is tangent to Ai Ai+1 , Ai Ai+2 (mod 3) (4) Bi is the second intersection point of Ci+1 , Ci+2 . circumcenters of triangles Ai Bi I are collinear.
Prove that
520. P moves on the incircle of tangential quadrilateral ABCD. P B, P D, the tangent at P cut AC at Q, R, T respectively. show that AT 2 AQ · AR = CT 2 CQ · CR 521. The perpendicular bisector of BC of 4ABC meets Γ its circumcircle with center O at P , Q, where A, P are on the same side of BC. Let I be the incenter of 4ABC and P I meets BC at R, QR meets Γ again at S, such that O, I, S are
51
collinear. ω that passes A, I, R meets P A at F . E lies on P R and QR = RE, EF meets Γ G, H (G lies within segment EF ). Prove that, 1 1 1 + = EF EH GE
52
Solutions 1. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763 2. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763 3. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763 4. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763 5. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763 6. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=20 7. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=20 8. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=20 9. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=20 10. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=20 11. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=20 12. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=40 13. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=40 14. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=40 15. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=40 16. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=40 17. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=40 18. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=40 19. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=60 20. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=60 21. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=60 22. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=60 23. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=60 24. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=80 25. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=80 26. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=80 27. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=80 28. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=80 29. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=80 30. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=100 31. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=100 32. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=120 33. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=120 53
34. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=120 35. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=120 36. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=120 37. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=120 38. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=140 39. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=140 40. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=140 41. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=140 42. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=140 43. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=140 44. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=140 45. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=160 46. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=160 47. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=160 48. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=160 49. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=160 50. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=160 51. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=180 52. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=180 53. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=180 54. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=180 55. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=180 56. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=180 57. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=200 58. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=200 59. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=200 60. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=200 61. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=200 62. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=200 63. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=200 64. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=200 65. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=200 66. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=200 67. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=200 68. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=200 54
69. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=220 70. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=220 71. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=220 72. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=220 73. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=220 74. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=220 75. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=220 76. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=220 77. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=220 78. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=220 79. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=220 80. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=220 81. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=220 82. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=220 83. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=240 84. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=240 85. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=240 86. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=240 87. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=240 88. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=240 89. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=240 90. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=260 91. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=260 92. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=260 93. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=260 94. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=260 95. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=260 96. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=260 97. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=260 98. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=260 99. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=260 100. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=260 101. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=260 102. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=260 103. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=280 55
104. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=280 105. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=280 106. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=280 107. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=280 108. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=280 109. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=280 110. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=280 111. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=280 112. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=280 113. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=300 114. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=300 115. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=300 116. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=300 117. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=300 118. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=300 119. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=300 120. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=300 121. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=300 122. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=300 123. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=300 124. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=300 125. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=300 126. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=300 127. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=300 128. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=320 129. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=320 130. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=320 131. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=320 132. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=320 133. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=320 134. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=320 135. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=320 136. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=340 137. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=340 138. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=340 56
139. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=340 140. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=340 141. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=340 142. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=340 143. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=340 144. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=340 145. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=340 146. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=360 147. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=360 148. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=360 149. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=360 150. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=360 151. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=360 152. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=360 153. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=360 154. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=360 155. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=360 156. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=360 157. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=360 158. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=360 159. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=360 160. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=360 161. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=380 162. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=380 163. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=380 164. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=380 165. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=380 166. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=380 167. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=380 168. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=380 169. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=380 170. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=400 171. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=400 172. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=400 173. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=400 57
174. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=400 175. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=400 176. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=400 177. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=400 178. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=420 179. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=420 180. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=420 181. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=420 182. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=420 183. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=420 184. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=420 185. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=420 186. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=420 187. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=420 188. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=420 189. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=420 190. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=420 191. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=440 192. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=440 193. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=440 194. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=440 195. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=440 196. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=440 197. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=440 198. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=440 199. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=440 200. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=440 201. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=440 202. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=440 203. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=440 204. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=440 205. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=460 206. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=460 207. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=460 208. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=460 58
209. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=460 210. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=460 211. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=460 212. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=460 213. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=460 214. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=460 215. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=460 216. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=460 217. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=460 218. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=480 219. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=480 220. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=480 221. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=480 222. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=480 223. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=480 224. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=480 225. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=500 226. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=500 227. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=500 228. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=500 229. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=500 230. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=500 231. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=500 232. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=500 233. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=500 234. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=520 235. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=520 236. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=520 237. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=520 238. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=520 239. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=520 240. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=520 241. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=520 242. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=520 243. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=520 59
244. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=540 245. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=540 246. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=540 247. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=540 248. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=540 249. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=540 250. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=540 251. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=540 252. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=540 253. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=540 254. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=560 255. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=560 256. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=560 257. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=560 258. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=560 259. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=560 259. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=560 260. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=560 261. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=560 262. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=560 263. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=560 264. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=580 265. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=580 266. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=580 267. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=580 268. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=580 269. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=580 270. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=580 271. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=580 272. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=580 273. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=580 274. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=580 275. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=580 275. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=600 276. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=600 60
277. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=600 278. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=600 279. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=600 280. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=600 281. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=600 282. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=600 283. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=600 284. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=600 285. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=600 286. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=620 287. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=620 288. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=620 289. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=620 290. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=620 291. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=620 292. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=620 293. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=620 294. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=620 295. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=640 296. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=640 297. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=640 298. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=640 299. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=640 300. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=640 301. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=640 302. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=640 303. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=640 304. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=640 305. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=660 306. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=660 307. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=660 308. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=660 309. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=660 310. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=660 311. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=660 61
312. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=660 313. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=660 314. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=660 315. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=660 316. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=680 317. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=680 318. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=680 319. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=680 320. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=680 321. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=680 322. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=680 323. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=680 324. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=680 325. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=700 326. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=700 327. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=700 328. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=700 329. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=700 330. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=700 331. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=720 332. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=720 333. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=720 334. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=720 335. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=720 336. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=720 337. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=720 338. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=720 339. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=720 340. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=720 341. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=720 342. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=740 343. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=740 344. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=740 345. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=740 346. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=740 62
347. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=740 348. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=740 349. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=740 350. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=740 351. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=760 352. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=760 353. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=760 354. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=760 355. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=760 356. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=760 357. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=760 358. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=760 359. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=780 360. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=780 361. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=780 362. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=780 363. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=780 364. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=780 365. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=800 366. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=800 367. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=800 368. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=800 369. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=800 370. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=800 371. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=800 372. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=820 373. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=820 374. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=820 375. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=820 376. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=820 377. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=820 377. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=840 378. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=840 379. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=840 380. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=840 63
381. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=840 382. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=840 383. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=840 384. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=860 385. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=860 386. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=860 387. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=860 387. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=860 388. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=860 389. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=860 390. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=860 391. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=860 392. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=880 393. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=880 394. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=880 395. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=880 396. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=880 397. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=880 397. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=880 398. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=880 399. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=880 400. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=880 401. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=900 402. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=900 403. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=900 404. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=900 405. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=900 406. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=900 407. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=920 408. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=920 409. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=920 410. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=920 411. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=920 412. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=920 413. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=920 64
414. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=920 415. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=920 416. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=920 417. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=940 418. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=940 419. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=940 420. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=940 421. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=940 422. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=940 423. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=960 424. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=960 425. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=960 426. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=960 427. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=960 428. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=960 429. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=960 430. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=960 431. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=980 432. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=960 433. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=960 434. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=960 435. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=960 436. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=960 437. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=960 438. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=960 439. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=980 440. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=980 441. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=980 442. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=980 443. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=980 444. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=980 445. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1020 446. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1020 447. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1020 448. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1020 65
449. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1020 450. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1020 451. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1040 452. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1020 453. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1020 454. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1020 455. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1020 456. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1020 457. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1020 458. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1020 459. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1020 460. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1040 461. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1020 462. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1020 463. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1020 464. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1020 465. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1020 466. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1020 467. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1020 468. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1040 469. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1040 470. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1040 471. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1040 472. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1040 473. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1040 474. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1040 475. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1040 476. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1060 477. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1060 478. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1060 479. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1060 480. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1060 481. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1060 482. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1060 483. http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1060 66
484. 485. 486. 487. 488. 489. 490. 491. 492. 493. 494. 495. 496. 497. 498. 499. 500. 501. 502. 503. 504. 505. 506. 507. 508. 509. 510. 511. 512. 513. 514. 515. 516. 517. 518. 519. 520. 521.
http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1080 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1080 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1080 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1080 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1080 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1080 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1100 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1100 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1100 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1100 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1100 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1100 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1100 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1100 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1120 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1120 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1120 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1120 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1120 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1120 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1120 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1120 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1140 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1140 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1140 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1140 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1140 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1140 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1140 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1140 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1160 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1160 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1160 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1160 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1160 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1160 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1160 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=331763&start=1160
67
