# APMO

## 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 …

## APMO Problems and Solutions 2018

Problem 1: Let H be the orthocenter of the triangle ABC. Let M and N be the midpoints of the sides AB and AC, respectively. Assume that H lies inside the quadrilateral BMNC and that the circumcircles of triangles BMH and CNH are tangent to each other. The line through H parallel to BC intersects …

## APMO Problems and Solutions 2019

Problem 1. Let ℤ+ be the set of positive integers. Determine all functions f : ℤ+→ ℤ+ such that a2 + f(a)f(b) is divisible by f(a) + b for all positive integers a and b. Problem 2. Let, m be a fixed positive integer. The infinite sequence {an}n≥1 is defined in the following way: a1 …

## 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 …