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HMMT 二月 2005 · GEN1 赛 · 第 4 题

HMMT February 2005 — GEN1 Round — Problem 4

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. Find all real solutions to x + ( x + 1) + ( x + 2) = ( x + 3) .
解析

英文解析

  1. Find all real solutions to x + ( x + 1) + ( x + 2) = ( x + 3) .
    Solution: 3
    3 2 3 2
    The equation simplifies to 3 x + 9 x + 15 x + 9 = x + 9 x + 27 x + 27, or equivalently,
    3 2 2
    2 x − 12 x − 18 = 2( x − 3)( x + 3 x + 3) = 0. The discriminant of x + 3 x + 3 is − 3 < 0,
    so the only real solution is x = 3. 1