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HMMT 十一月 2013 · GEN 赛 · 第 7 题

HMMT November 2013 — GEN Round — Problem 7

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 6 ] Find the largest real number λ such that a + b + c + d ≥ ab + λbc + cd for all real numbers a, b, c, d .
解析
  1. [ 6 ] Find the largest real number λ such that a + b + c + d ≥ ab + λbc + cd for all real numbers a, b, c, d . 3 2 2 2 2 Answer: Let f ( a, b, c, d ) = ( a + b + c + d ) − ( ab + λbc + cd ). For fixed ( b, c, d ), f is minimized 2 b c at a = , and for fixed ( a, b, c ), f is minimized at d = , so simply we want the largest λ such that 2 2 b c 3 2 2 3 3 f ( , b, c, ) = ( b + c ) − λbc is always nonnegative. By AM-GM, this holds if and only if λ ≤ 2 = . 2 2 4 4 2