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Problem 1046

AMC 12 late, AIME early

Algebra Difficulty 4.7 Prove it Mathematical Olympiad in Ukraine · Ukraine

Let $a, b \in [-1,1]$ . Prove that $a\sqrt{1-b^2} + b\sqrt{1-a^2} \le 1$ .

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

By substitution: $a = \sin \alpha$ , $b = \sin \beta$ the given inequality becomes: $\sin \alpha \cos \beta + \sin \beta \cos \alpha \le 1$ . This inequality holds for an arbitrary $\alpha, \beta$ .

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.