HMMT 二月 2011 · 冲刺赛 · 第 13 题

HMMT February 2011 — Guts Round — Problem 13

[ 8 ] Let a , b , and c be the side lengths of a triangle, and assume that a ≤ b and a ≤ c . Let x = . 2 ax If r and R denote the inradius and circumradius, respectively, find the minimum value of . rR

解析

[ 8 ] Let a , b , and c be the side lengths of a triangle, and assume that a ≤ b and a ≤ c . Let x = . 2 ax If r and R denote the inradius and circumradius, respectively, find the minimum value of . rR Answer: 3 r ( a + b + c ) r ( a + b + c ) abc abc It is well-known that both and are equal to the area of triangle ABC . Thus = , 4 R 2 4 R 2 and abc Rr = . 2( a + b + c ) Guts Round 2 a Since a ≤ b and a ≤ c , we have ≤ 1. We thus obtain that bc ax a ( b + c − a ) / 2 = abc rR 2( a + b + c ) ( a + b + c )( b + c − a ) = bc 2 2 ( b + c ) − a = bc 2 2 ( b + c ) a = − bc bc 2 b c a = + + 2 − c b bc b c ≥ + + 2 − 1 c b ≥ 2 + 2 − 1 = 3 Equality is achieved when a = b = c .