Home > Olympiad > IMO 2015 Problem 1

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

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