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## IMO 2015 Problem 1

Problem 1. We say that a finite set ${\mathcal{S}}$ of points in the plane is balanced if, for any two different points ${A}$ and ${B}$ in ${\mathcal{S}}$, there is a point ${C}$ in ${\mathcal{S}}$ such that ${AC=BC}$. We say that ${\mathcal{S}}$ is center-free if for any three different points ${A}$, ${B}$ and ${C}$ in ${\mathcal{S}}$, there is no points ${P}$ in ${\mathcal{S}}$ such that ${PA=PB=PC}$.

(a) Show that for all integers ${n\ge 3}$, there exists a balanced set having ${n}$ points.

(b) Determine all integers ${n\ge 3}$ for which there exists a balanced center-free set having ${n}$ points.

Problem 2. Find all triples of positive integers ${(a, b, c)}$ such that ${ab-c, bc-a, ca-b}$ are all powers of 2.

Problem 3. Let ${ABC}$ be an acute triangle with ${AB > AC}$. Let ${\Gamma }$ be its cirumcircle., ${H}$ its orthocenter, and ${F}$ the foot of the altitude from ${A}$. Let ${M}$ be the midpoint of ${BC}$. Let ${Q }$ be the point on ${ \Gamma }$ such that ${\angle HQA }$ and let ${K }$ be the point on ${\Gamma }$ such that ${\angle HKQ }$. Assume that the points ${A,B,C,K }$and ${Q }$ are all different and lie on ${\Gamma}$ in this order.

Prove that the circumcircles of triangles ${KQH }$ and ${FKM }$ are tangent to each other.

Source: AoPS