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Prove the existence of the configuration and ...

Source: Balkan MO ShortList 2009 A2

April 6, 2020

Problem Statement

Let ABCDABCD be a square and points MM ∈\in BCBC, N∈CDN \in CD, PP ∈\in DADA, such that ∠BAM\angle BAM == xx, ∠CMN\angle CMN == 2x2x, ∠DNP\angle DNP == 3x3x
[*] Show that, for any x∈(0,π8)x \in (0, \tfrac{\pi}{8} ), such a configuration exists [*] Determine the number of angles x∈(0,π8)x \in ( 0, \tfrac{\pi}{8} ) for which ∠APB=4x\angle APB =4x
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