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INMO 2019 P2

Source:

January 20, 2019
combinatorics

Problem Statement

Let A1B1C1D1E1A_1B_1C_1D_1E_1 be a regular pentagon.For 2≤n≤11 2 \le n \le 11, let AnBnCnDnEnA_nB_nC_nD_nE_n be the pentagon whose vertices are the midpoint of the sides An−1Bn−1Cn−1Dn−1En−1A_{n-1}B_{n-1}C_{n-1}D_{n-1}E_{n-1}. All the 55 vertices of each of the 1111 pentagons are arbitrarily coloured red or blue. Prove that four points among these 5555 points have the same colour and form the vertices of a cyclic quadrilateral.
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