HMMT 二月 2011 · 冲刺赛 · 第 13 题

英文解析

13. [ 8 ] Let a , b , and c be the side lengths of a triangle, and assume that a ≤ b and a ≤ c . Let x = .
    ax 2
    If r and R denote the inradius and circumradius, respectively, find the minimum value of .
    Answer: 3 r rr ( 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
    abcand
    Rr = .
    2( a + b + c )
    Guts Rounda 2
    Since a ≤ b and a ≤ c , we have ≤ 1. We thus obtain thatbcax a ( b + c − a ) / 2
    abc=
    2( a + b + c )r r
    ( a + b + c )( b + c − a )
    bc=
    2 2
    ( b + c ) − abc=
    2 2
    ( b + c ) a = −
    bc bcb c a 2 = + + 2 −
    c b bcb c
    ≥ + + 2 − 1
    c b
    ≥ 2 + 2 − 1 = 3
    Equality is achieved when a = b = c .