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HMMT 十一月 2011 · GEN 赛 · 第 5 题

HMMT November 2011 — GEN Round — Problem 5

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 5 ] Find all real values of x for which
    1 1 1
    √ + √ = .
    √ √
    x + x − 2 x + 2 + x 4
解析

英文解析

  1. [ 5 ] Find all real values of x for which
    1 1 1
    √ √
    √ + √ = .
    x + x − 2 x + 2 + x 4
    Answer: We note that 257
    1 1 116
    √ √
    = +
    √ √
    x + x − 2 x + 2 + x 4
    √ √
    √ √
    x − x − 2 x + 2 − x = √ √ + √ √
    √ √ √ √
    ( x + x − 2)( x − x − 2) ( x + 2 + x )( x + 2 − x )
    √ √
    √ √
    x − x − 2 x + 2 − x = +
    2 2
    √ √
    = ( x + 2 − x − 2) ,1
    so that 2
    √ √
    2 x + 2 − 2 x − 2 = 1 .
    Squaring, we get that
    √ √
    8 x − 8 ( x + 2)( x − 2) = 1 ⇒ 8 x − 1 = 8 ( x + 2)( x − 2) .
    Squaring again gives
    2 2
    64 x − 16 x + 1 = 64 x − 256 ,
    so we get that x = .257
    16