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HMMT 二月 2011 · 冲刺赛 · 第 2 题

HMMT February 2011 — Guts Round — Problem 2

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 4 ] Let a ? b = sin a cos b for all real numbers a and b . If x and y are real numbers such thatx ? y − y ? x = 1, what is the maximum value of x ? y + y ? x ?
解析

英文解析

  1. [ 4 ] Let a ? b = sin a cos b for all real numbers a and b . If x and y are real numbers such thatx ? y − y ? x = 1, what is the maximum value of x ? y + y ? x ?
    Answer: 1
    We have x ? y + y ? x = sin x cos y + cos x sin y = sin( x + y ) ≤ 1. Equality is achieved when x = andπy = 0. Indeed, for these values of x and y , we have x ? y − y ? x = sin x cos y − cos x sin y = sin( x − y ) =2
    sin = 1.π 2