# 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}n1 is defined in the following way: a1 is a positive integer, and for every integer n 1 we have

For each m, determine all possible values of a1 such that every term in the sequence is an integer.

Problem 3. Let ABC be a scalene triangle with circumcircle Γ. Let, M be the midpoint of BC. A variable point P is selected in the line segment AM. The circumcircles of triangles BPM and CPM intersect Γ again at points D and E, respectively. The lines DP and EP intersect (a second time) the circumcircles to triangles CPM and BPM at X and Y, respectively. Prove that as P varies, the circumcircle of 4AXY passes through a fixed-point T distinct from A.

Problem 4. Consider a 2018 × 2019 board with integers in each unit square. Two-unit squares are said to be neighbours if they share a common edge. In each turn, you choose some unit squares. Then for each chosen unit square the average of all its neighbours are calculated. Finally, after these calculations are done, the number in each chosen unit square is replaced by the corresponding average. Is it always possible to make the numbers in all squares become the same after finitely many turns?

Problem 5. Determine all the functions f : ℝ → ℝ such that
f(x2 + f(y)) = f(f(x)) + f(y2) + 2f(xy)
for all real numbers x and y.