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Collinearity of circumcentres

Source: Canada 2001

March 4, 2006
geometrycircumcirclerotationgeometric transformationhomothetyperpendicular bisectorgeometry unsolved

Problem Statement

Let P0P_0, P1P_1, P2P_2 be three points on the circumference of a circle with radius 11, where P1P2=t<2P_1P_2 = t < 2. For each i≥3i \ge 3, define PiP_i to be the centre of the circumcircle of △Pi−1Pi−2Pi−3\triangle P_{i-1} P_{i-2} P_{i-3}. (1) Prove that the points P1,P5,P9,P13,⋯P_1, P_5, P_9, P_{13},\cdots are collinear. (2) Let xx be the distance from P1P_1 to P1001P_{1001}, and let yy be the distance from P1001P_{1001} to P2001P_{2001}. Determine all values of tt for which xy500\sqrt[500]{ \frac xy} is an integer.
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