HMMT 二月 2008 · GEN2 赛 · 第 4 题

HMMT February 2008 — GEN2 Round — Problem 4

[ 3 ] Suppose that a, b, c, d are real numbers satisfying a ≥ b ≥ c ≥ d ≥ 0, a + d = 1, b + c = 1, and ac + bd = 1 / 3. Find the value of ab − cd .

解析

[ 3 ] Suppose that a, b, c, d are real numbers satisfying a ≥ b ≥ c ≥ d ≥ 0, a + d = 1, b + c = 1, and ac + bd = 1 / 3. Find the value of ab − cd . √ 2 2 Answer: We have 3 ( ) 2 1 8 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 = . 3 Comment: Another way to solve this problem is to use the trigonometric substitutions a = sin θ , b = sin φ , c = cos φ , d = cos θ .