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IMO ShortList 1982 C2

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September 10, 2010
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Problem Statement

Let ABCDABCD be a convex plane quadrilateral and let A1A_1 denote the circumcenter of △BCD\triangle BCD. Define B1,C1,D1B_1, C_1,D_1 in a corresponding way.
(a) Prove that either all of A1,B1,C1,D1A_1,B_1, C_1,D_1 coincide in one point, or they are all distinct. Assuming the latter case, show that A1A_1, C1 are on opposite sides of the line B1D1B_1D_1, and similarly,B1,D1 B_1,D_1 are on opposite sides of the line A1C1A_1C_1. (This establishes the convexity of the quadrilateral A1B1C1D1A_1B_1C_1D_1.)
(b) Denote by A2A_2 the circumcenter of B1C1D1B_1C_1D_1, and define B2,C2,D2B_2, C_2,D_2 in an analogous way. Show that the quadrilateral A2B2C2D2A_2B_2C_2D_2 is similar to the quadrilateral ABCD.ABCD.
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