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Found problems: 1

Ukraine Correspondence MO - geometry, 2006.10
Tags: geometry , mixtilinear excircle , radiii , fixed , isosceles , Ukraine Correspondence
Let $ABC$ be an isosceles triangle ($AB=AC$). An arbitrary point $M$ is chosen on the extension of the $BC$ beyond point $B$. Prove that the sum of the radius of the circle inscribed in the triangle $AM​​B$ and the radius of the circle tangent to the side $AC$ and the extensions of the sides $AM, CM$ of the triangle $AMC$ does not depend on the choice of point $M$.