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HMMT 十一月 2014 · THM 赛 · 第 8 题

HMMT November 2014 — THM 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. Consider the parabola consisting of the points ( x, y ) in the real plane satisfying
    ( y + x ) = ( y − x ) + 3( y − x ) + 3 .2
    Find the minimum possible value of y .
解析

英文解析

  1. Consider the parabola consisting of the points ( x, y ) in the real plane satisfying
    ( y + x ) = ( y − x ) + 3( y − x ) + 3 .2
    Find the minimum possible value of y .
    Answer: − Let w = y − x . Adding w to both sides and dividing by two gives 1
    2 22
    w + 4 w + 3 ( w + 2) − 1
    y = = ,
    2 2
    which is minimized when w = − 2. This yields y = − .1 2