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\sin a _A \sin a _B \sina _C\leq \frac{3\sqrt{6}}{32}

Source: BMO Shortlist 2015 G1 (Romania)

September 27, 2018
geometrycircumcircletrigonometry

Problem Statement

In an acute angled triangle ABCABC , let BB′BB' and CC′CC' be the altitudes. Ray C′B′C'B' intersects the circumcircle at B′′B'' andl let αA\alpha_A be the angle ABB′′^\widehat{ABB''}. Similarly are defined the angles αB\alpha_B and αC\alpha_C. Prove that sin⁡αAsin⁡αBsin⁡αC≤3632\displaystyle\sin \alpha _A \sin \alpha _B \sin \alpha _C\leq \frac{3\sqrt{6}}{32} (Romania)
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