Maths Olympiad Prep

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Algebra Difficulty 5.9 AIME, harder Prove it

Example 3 In △ABC\triangle A B C, prove:
sin⁡A+sin⁡B+sin⁡C⩽323\sin A+\sin B+\sin C \leqslant \frac{3}{2} \sqrt{3}

Solution

Prove: Let f(x)=−sin⁡x(0⩽x<π)f(x)=-\sin x \quad(0 \leqslant x<\pi), according to Jensen's inequality:
−sin⁡A+B+C3⩽13(−sin⁡A−sin⁡B−sin⁡C)-\sin \frac{A+B+C}{3} \leqslant \frac{1}{3}(-\sin A-\sin B-\sin C)
i.e., sin⁡A+sin⁡B+sin⁡C⩽323\sin A+\sin B+\sin C \leqslant \frac{3}{2} \sqrt{3}.
Equality holds if and only if A=B=C=π3A=B=C=\frac{\pi}{3}.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.