OLYMPIAD 2018. Task 1. Let f (x) = ln(1 − x2 ). Find f (6) (0) (the value of the sixth derivative of f (x) at x = 0). Task 2. Two pedestrians start walking towards each other from destinations A and B with constant speeds. The speed of the first pedestrian (who started in A) is greater than the speed of the second one. When the first pedestrian had passed a half of the path, the second one had 8 miles left. When the second pedestrian had passed a half of the path, the first one had 3 miles left. Find the distance between A and B. Task 3. The base of the regular pyramid is a polygon with an odd number of sides. Is it possible to define a direction of each edge in such a way that the sum of these vectors will be equal to 0? √ √ √ √ √ Task 4. Which number is larger: 2 + 3 3 + 2 − 3 3, or 2 3 2? Task 5. Given two non-scalar 2×2 matrices A and B such that A2 = αI, B 2 = βI, where α and β are scalars, and I is the identity matrix, prove that A·B+B·A = γI, where γ is some scalar. Task 6. A function f (x) = ax2 + bx + c satisfies a condition |f (x)| ≤ 1 for all |x| ≤ 1. Find the maximum possible absolute value of the derivative f ′ (x) of this function on the segment [−1, 1]. Task 7. A sequence {xn } is defined recursively by formula: xn+1 = ln(1 + xn ), where x1 is a real number from the interval (0, 1). Find the limit lim nx1n . n→∞ (2018) (2017) (2016) (n ) Task 8. Find the value of 0 − 1 + 2 − . . . , where k is defined as n! if 0 ≤ k ≤ n, and 0 otherwise. (n−k)!k! Task 9. Let P (x1 , x2 , x3 ) and Q(x1 , x2 , x3 ) be some polynomials with • a) complex • b) real coefficients. Is it possible they satisfy the identity P 2 + Q2 = x21 + x22 + x23 ? Task 10. There are n cities in the country. Two travelers visit each of n cities. The first traveler always picks the farthest city he had not visited before, and the second traveler always picks the nearest one. Prove that the total length of the path of the first traveler will not be shorter than the total length of the path of the second one. 1