APMO Problems and Solutions 2017

Problem 1: We call a 5-tuple of integers arrangeable if its elements can be labeled a, b, c, d, e in some order so that a – b + c – d + e = 29. Determine all 2017-tuples of integers n1, n2, …, n2017such that if we place them in a circle in clockwise order, then any 5-tuple of numbers in consecutive positions on the circle is arrangeable.

Problem 2. Let ABC be a triangle with AB < AC. Let D be the intersection point of the internal bisector of angle BAC and the circumcircle of ABC. Let Z be the intersection point of the perpendicular bisector of AC with the external bisector of angle ∆BAC. Prove that the midpoint of the segment AB lies on the circumcircle of triangle ADZ.

Problem 3: Let A(n) denote the number of sequences a1 ≥ a2 ≥ … ≥ ak of positive integers for which a1 +· · · + ak = n and each ai + 1 is a power of two (i = 1, 2, …, k).
Let B(n) denote the number of sequences b1 ≥ b2 ≥ … ≥ bm of positive integers for which b1 + · · · + bm = n and each inequality bj ≥ 2bj+1 holds (j = 1, 2, …, m – 1).
Prove that A(n) = B(n) for every positive integer n.

Problem 4. Call a rational number r powerful if r can be expressed in the form pk/q for some relatively prime positive integers p, q and some integer k > 1. Let a, b, c be positive rational numbers such that abc = 1. Suppose there exist positive integers x, y, z such that ax + by + cz is an integer. Prove that a, b, c is all powerful.

Problem 5. Let n be a positive integer. A pair of n-tuples (a1, … an) and (b1, …, bn) with integer entries is called an exquisite pair if |a1b1 + … + anbn| ≤ 1.
Determine the maximum number of distinct n-tuples with integer entries such that any two of them form an exquisite pair.