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HMMT 二月 2008 · GEN2 赛 · 第 4 题

HMMT February 2008 — GEN2 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. [ 3 ] Suppose that a, b, c, d are real numbers satisfying a ≥ b ≥ c ≥ d ≥ 0, a + d = 1, b + c = 1, andac + bd = 1 / 3. Find the value of ab − cd .
解析

英文解析

  1. [ 3 ] Suppose that a, b, c, d are real numbers satisfying a ≥ b ≥ c ≥ d ≥ 0, a + d = 1, b + c = 1, andac + bd = 1 / 3. Find the value of ab − cd .
    √
    2 2
    Answer: We have
    ( )3
    1 82
    2 2 2 2 2 2
    ( ab − cd ) = ( a + d )( b + c ) − ( ac + bd ) = (1)(1) − = .
    3 9
    √
    2 2
    Since a ≥ b ≥ c ≥ d ≥ 0, ab − cd ≥ 0, so ab − cd = .
    Comment: Another way to solve this problem is to use the trigonometric substitutions a = sin θ ,3
    b = sin φ , c = cos φ , d = cos θ .