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HMMT 二月 2004 · 冲刺赛 · 第 3 题

HMMT February 2004 — Guts Round — Problem 3

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

  1. [5] If the system of equations
    | x + y | = 99
    | x − y | = chas exactly two real solutions ( x, y ), find the value of c .
    HARVARD-MIT MATHEMATICS TOURNAMENT, FEBRUARY 28, 2004 — GUTS ROUND

英文原题

[5] If the system of equations
| x + y | = 99
| x − y | = c
has exactly two real solutions ( x, y ), find the value of c .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
HARVARD-MIT MATHEMATICS TOURNAMENT, FEBRUARY 28, 2004 — GUTS ROUND

解析

英文解析

  1. If the system of equations
    | x + y | = 99
    | x − y | = chas exactly two real solutions ( x, y ), find the value of c .
    Solution: 0
    If c < 0, there are no solutions. If c > 0 then we have four possible systems of linearequations given by x + y = ± 99, x − y = ± c , giving four solutions ( x, y ). So we musthave c = 0, and then we do get two solutions ( x = y , so they must both equal ± 99 / 2).