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HMMT 二月 2004 · 冲刺赛 · 第 3 题

HMMT February 2004 — Guts Round — Problem 3

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [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
  1. If the system of equations | x + y | = 99 | x − y | = c has 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 linear equations given by x + y = ± 99, x − y = ± c , giving four solutions ( x, y ). So we must have c = 0, and then we do get two solutions ( x = y , so they must both equal ± 99 / 2).