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HMMT 十一月 2014 · GEN 赛 · 第 8 题

HMMT November 2014 — GEN Round — Problem 8

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. Let a , b , c , x be reals with ( a + b )( b + c )( c + a ) 6 = 0 that satisfy
    2 2 2 2 2 2
    a a b b c c = + 20 , = + 14 , and = + x.
    a + b a + c b + c b + a c + a c + b
    Compute x .
解析

英文解析

  1. Let a , b , c , x be reals with ( a + b )( b + c )( c + a ) 6 = 0 that satisfy
    2 2 2 2 2 2
    a a b b c c = + 20 , = + 14 , and = + x.
    a + b a + c b + c b + a c + a c + b
    Compute x .
    General Test
    Answer: − 34 Note that
    2 2 2 2 2 2 2 2 2 2 2 2
    a b c a b c a − b b − c c − a + + − − − = + +
    a + b b + c c + a c + a a + b b + c a + b b + c c + a = ( a − b ) + ( b − c ) + ( c − a ) = 0 .
    Thus, when we sum up all the given equations, we get that
    20 + 14 + x = 0. Therefore, x = − 34.