# 2019 IMO

## Problem 1

Let be the set of integers. Determine all functions such that, for all integers and ,

## Problem 2

In triangle , point lies on side and point lies on side . Let and be points on segments and , respectively, such that is parallel to . Let be a point on line , such that lies strictly between and , and . Similarly, let be the point on line , such that lies strictly between and , and .

Prove that points , and are concyclic.

## Problem 3

A social network has users, some pairs of whom are friends. Whenever user is friends with user , user is also friends with user . Events of the following kind may happen repeatedly, one at a time: Three users , , and such that is friends with both and , but and are not friends, change their friendship statuses such that and are now friends, but is no longer friends with , and no longer friends with . All other friendship statuses are unchanged. Initially, users have friends each, and users have friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.

## Problem 4

Find all pairs of positive integers such that

## Problem 5

The Bank of Bath issues coins with an on one side and a on the other. Harry has of these coins arranged in a line from left to right. He repeatedly performs the following operation:

If there are exactly coins showing , then he turns over the coin from the left; otherwise, all coins show and he stops. For example, if the process starting with the configuration would be , which stops after three operations.

(a) Show that, for each initial configuration, Harry stops after a finite number of operations.

(b) For each initial configuration , let be the number of operations before Harry stops. For example, and . Determine the average value of over all possible initial configurations .

## Problem 6

Let be the incenter of acute triangle with . The incircle of is tangent to sides , , and at , , and , respectively. The line through perpendicular to meets ω again at . Line meets ω again at . The circumcircles of triangles and meet again at . Prove that lines and meet on the line through perpendicular to .