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< A_1B_1C_1=? , <QOP = 1/2 <B_1C_1D_1 , equal rhombuses ABCD, A_1B_1C_1D_1

Source: 2008 Moldova JBMO TST 1.3

July 12, 2020
geometryrhombuscongruentangles

Problem Statement

Rhombuses ABCDABCD and A1B1C1D1A_1B_1C_1D_1 are equal. Side BCBC intersects sides B1C1B_1C_1 and C1D1C_1D_1 at points MM and NN respectively. Side ADAD intersects sides A1B1A_1B_1 and A1D1A_1D_1 at points QQ and PP respectively. Let OO be the intersection point of lines MPMP and QNQN. Find ∠A1B1C1\angle A_1B_1C_1 , if ∠QOP=12∠B1C1D1\angle QOP = \frac12 \angle B_1C_1D_1.
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