Track / Stage 5 / 153 of 400 #753 of 1964

Problem 753

AIME late

Algebra Difficulty 5.3 Prove it

a) $1<\cos \alpha+\cos \beta+\cos \gamma \leq 3 / 2$

b) $1<\sin (\alpha / 2)+\sin (\beta / 2)+\sin (\gamma / 2) \leq 3 / 2$ .

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

a) According to problem $\underline{12.38} \cos \alpha+\cos \beta+\cos \gamma=(R+r) / R$ . In addition, $r \leq R / 2$ (problem $\underline{10.26}$ ).

b) Follows from a) (see remark).

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