Track / Stage 5 / 209 of 400 #809 of 1964
Problem 809 AIME late Difficulty 5.3 Prove it
a) 1 < cos α + cos β + cos γ ≤ 3 / 2 1<\cos \alpha+\cos \beta+\cos \gamma \leq 3 / 2 1 < cos α + cos β + cos γ ≤ 3/2
b) 1 < sin ( α / 2 ) + sin ( β / 2 ) + sin ( γ / 2 ) ≤ 3 / 2 1<\sin (\alpha / 2)+\sin (\beta / 2)+\sin (\gamma / 2) \leq 3 / 2 1 < sin ( α /2 ) + sin ( β /2 ) + sin ( γ /2 ) ≤ 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.
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Official solution a) According to problem 12.38 ‾ cos α + cos β + cos γ = ( R + r ) / R \underline{12.38} \cos \alpha+\cos \beta+\cos \gamma=(R+r) / R 12.38 cos α + cos β + cos γ = ( R + r ) / R r ≤ R / 2 r \leq R / 2 r ≤ R /2 10.26 ‾ \underline{10.26} 10.26
b) Follows from a) (see remark).
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Source: NuminaMath-1.5 ,
licensed Apache-2.0 .
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