Track / Stage 5 / 4 of 400 #1244 of 2604

Problem 1244

AIME late

Geometry Difficulty 5.0 Prove it BMO Round 1 · United Kingdom · 2009

The obtuse-angled triangle $ABC$ has sides of length $a$ , $b$ and $c$ opposite the angles $\angle A$ , $\angle B$ and $\angle C$ respectively. Prove that
$a^3 \cos A + b^3 \cos B + c^3 \cos C < abc.$

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.

We don't reproduce this publisher's solutions. Their own solution is here — work the problem first.

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.