IMO Shortlist 1999 problem G4

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Dodao/la: arhiva
2. travnja 2012.
For a triangle T = ABC we take the point X on the side (AB) such that AX/AB=4/5, the point Y on the segment (CX) such that CY = 2YX and, if possible, the point Z on the ray (CA such that \widehat{CXZ} = 180 - \widehat{ABC}. We denote by \Sigma the set of all triangles T for which
\widehat{XYZ} = 45. Prove that all triangles from \Sigma are similar and find the measure of their smallest angle.
Izvor: Međunarodna matematička olimpijada, shortlist 1999