## IMO 2015 Problem 1

**Problem 1.** We say that a finite set of points in the plane is *balanced* if, for any two different points and in , there is a point in such that . We say that is *center-free* if for any three different points , and in , there is no points in such that .

(a) Show that for all integers , there exists a balanced set having points.

(b) Determine all integers for which there exists a balanced center-free set having points.

**Problem 2.** Find all triples of positive integers such that are all powers of 2.

**Problem 3.** Let be an acute triangle with . Let be its cirumcircle., 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.

*Source: AoPS*