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Problem 1484

National Olympiad, first round

Algebra Difficulty 6.0 Prove it BMO Round 1 · United Kingdom · 2014

The angles $A$ , $B$ and $C$ of a triangle are measured in degrees, and the lengths of the opposite sides are $a$ , $b$ and $c$ respectively. Prove that
$60 \le \frac{aA + bB + cC}{a+b+c} < 90.$

This one wants a proof. Work it on paper, then check yourself against the publisher's own solution, linked below. Be honest about it: the record is only any use to you if it is.

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Source: UK Mathematics Trust, licensed © UK Mathematics Trust; question papers published free at bmos.ukmt.org.uk. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project. Solutions are the publisher's, linked not copied.