# 2015 IMO Problems

## Problem 1

We say that a finite set in the plane is * balanced *
if, for any two different points , in , there is
a point in such that . We say that
is *centre-free* if for any three points , , in
, there is no point in such that
.

- Show that for all integers , there exists a balanced set consisting of points.
- Determine all integers for which there exists a balanced centre-free set consisting of points.

## Problem 2

Determine all triples of positive integers such that each of the numbers is a power of 2.

(*A power of 2 is an integer of the form where is a non-negative integer *).

## Problem 3

Let be an acute triangle with . Let be its circumcircle, its orthocenter, and the foot of the altitude from . Let be the midpoint of . Let be the point on such that and let be the point on such that . Assume that the points , , , and are all different and lie on in this order.

Prove that the circumcircles of triangles and are tangent to each other.

## Problem 4

Triangle has circumcircle and circumcenter . A circle with center intersects the segment at points and , such that , , , and are all different and lie on line in this order. Let and be the points of intersection of and , such that , , , , and lie on in this order. Let be the second point of intersection of the circumcircle of triangle and the segment . Let be the second point of intersection of the circumcircle of triangle and the segment .

Suppose that the lines and are different and intersect at the point . Prove that lies on the line .

## Problem 5

Let be the set of real numbers. Determine all functions satisfying the equation

for all real numbers and .

## Problem 6

The sequence of integers satisfies the conditions:

(i) for all ,

(ii) for all .

Prove that there exist two positive integers and for whichfor all integers and such that .