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Found problems: 1

2024 Euler Olympiad, Round 2, 3
Tags: geometry , Quadralateral , Euler
Consider a convex quadrilateral \(ABCD\) with \(AC > BD\). In the plane of this quadrilateral, points \(M\) and \(N\) are chosen such that triangles \(ABM\) and \(CDN\) are equilateral, and segments \(MD\) and \(NA\) intersect lines \(AB\) and \(CD\) respectively. Similarly, points \(P\) and \(Q\) are chosen such that triangles \(ADP\) and \(BCQ\) are equilateral, but here segments \(PB\) and \(QA\) do not intersect lines \(AD\) and \(BC\) respectively. Prove that \(MN = AC + BD\) if and only if \(PQ = AC - BD\). [i]Proposed by Zaza Meliqidze, Georgia [/i]