IMO 2016 Shortlist G3
Let B = (−1,0) and C = (1,0) be fixed points on the coordinate plane. A nonempty, bounded subset S of the plane is said ...
Category: Geometry
Problem
Let B = (−1,0) and C = (1,0) be fixed points on the coordinate plane. A nonempty, bounded subset S of the plane is said to be nice if (i) there is a point T in S such that for every point Q in S, the segment TQ lies entirely in S; and (ii) for any triangle P1P2P3, there exists a unique point A in S and a permutation σ of the indices {1,2,3} for which triangles ABC and Pσ(1)Pσ(2)Pσ(3) are similar. Prove that there exist two distinct nice subsets S and S0 of the set {(x,y) : x ⩾ 0,y ⩾ 0} such that if A ∈ S and A0 ∈ S0 are the unique choices of points in (ii), then the product BA · BA0 is a constant independent of the triangle P1P2P3.