Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.0 AIME, harder Find the answer

5. Suppose $a, b$ , and $c$ are the sides of a triangle opposite angles $\alpha, \beta$ , and $\gamma$ respectively. If $\cos \beta<0$ and $\cos \alpha<\cos \gamma$ , arrange $a, b$ , and $c$ in increasing order.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Answer: $c, a, b$
Solution: $\beta$ must be obtuse and therefore the largest angle, and so $b$ is the longest side. As for $a$ and $c$ , since $\alpha$ and $\gamma$ must both be acute, cos is decreasing and thus $\alpha>\gamma$ , so $a>c$ .

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