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HMMT 二月 2002 · ADV 赛 · 第 9 题

HMMT February 2002 — ADV Round — Problem 9

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. Given that a, b, c are positive real numbers and log b + log c + log a = 0, find the valuea b c
    3 3 3
    of (log b ) + (log c ) + (log a ) .
    a b c
解析

英文解析

  1. Given that a, b, c are positive real numbers and log b + log c + log a = 0, find thea b c
    3 3 3
    value of (log b ) + (log c ) + (log a ) .
    a b c
    Solution: 3 . Let x = log b and y = log c ; then log a = − ( x + y ). Thus we want to a b c
    3 3 3 2 2
    compute the value of x + y − ( x + y ) = − 3 x y − 3 xy = − 3 xy ( x + y ). On the other hand,
    − xy ( x + y ) = (log b )(log c )(log a ) = 1, so the answer is 3.
    a b c