APMO Problems and Solutions 2020

Problem 1. Let Γ be the circumcircle of ∆ABC. Let D be a point on the side BC. The tangent to Γ at A intersects the parallel line to BA through D at point E. The segment CE intersects Γ again at F. Suppose B, D, F, E are concyclic. Prove that AC, BF, DE are concurrent.

Problem 2. Show that r = 2 is the largest real number r which satisfies the following condition: If a sequence a1, a2, … of positive integers fulfills the inequalities

Math Olympiad problems
for every positive integer n, then there exists a positive integer M such that an+2 = an for every n ≥ M.

Problem 3. Determine all positive integers k for which there exist a positive integer m and a set S of positive integers such that any integer n > m can be written as a sum of distinct elements of S in exactly k ways.

Problem 4. Let Z denote the set of all integers. Find all polynomials P(x) with integer coefficients that satisfy the following property:
For any infinite sequence a1, a2, … of integers in which each integer in Z appears exactly once, there exist indices i < j and an integer k such that ai + ai+1 + · · · + aj = P(k).

Problem 5. Let n ≥ 3 be a fixed integer. The number 1 is written n times on a blackboard. Below the blackboard, there are two buckets that are initially empty. A move consists of erasing two of the numbers a and b, replacing them with the numbers 1 and a + b, then adding one stone to the first bucket and gcd(a, b) stones to the second bucket. After some finite number of moves, there are s stones in the first bucket and t stones in the second bucket, where s and t are positive integers. Find all possible values of the ratio t/s.