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hard inequality: sin < CAM + sin < CBM <= 2 / sqrt(3)

Source: Bulgaria 97

August 5, 2004
inequalitiestrigonometrygeometryrectangleanalytic geometrygeometry proposed

Problem Statement

I couldn't solve this problem and the only solution I was able to find was very unnatural (it was an official solution, I think) and I couldn't be satisfied with it, so I ask you if you can find some different solutions. The problem is really great one! If MM is the centroid of a triangle ABCABC, prove that the following inequality holds: sin⁡∠CAM+sin⁡∠CBM≤23.\sin\angle CAM+\sin\angle CBM\leq\frac{2}{\sqrt3}. The equality occurs in a very strange case, I don't remember it.
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