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Genius strikes again: nobody solved this in contest!

Source: Kyiv City MO 2024 Round 2, Problem 7.4

February 4, 2024
geometry

Problem Statement

Points XX and YY are chosen inside an acute-angled triangle ABCABC with altitude ADAD so that ∠BXA+∠ACB=180∘,∠CYA+∠ABC=180∘\angle BXA + \angle ACB = 180^\circ , \angle CYA + \angle ABC = 180^\circ, and CD+AY=BD+AXCD + AY = BD + AX. Point MM is chosen on the ray BXBX so that XX lies on segment BMBM and XM=ACXM = AC, and point NN is chosen on the ray CYCY so that YY lies on segment CNCN and YN=ABYN = AB. Prove that AM=ANAM = AN.
Proposed by Mykhailo Shtandenko
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