返回题库

HMMT 二月 2005 · 代数 · 第 3 题

HMMT February 2005 — Algebra — Problem 3

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Let x , y , and z be distinct real numbers that sum to 0. Find the maximum possible value of xy + yz + zx . 2 2 2 x + y + z ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ a + b b + c c + a
解析
  1. Let x , y , and z be distinct real numbers that sum to 0. Find the maximum possible value of xy + yz + zx . 2 2 2 x + y + z Solution: − 1 / 2 2 2 2 2 Note that 0 = ( x + y + z ) = x + y + z + 2 xy + 2 yz + 2 zx . Rearranging, we get that 1 2 2 2 xy + yz + zx = − ( x + y + z ), so that in fact the quantity is always equal to − 1 / 2. 2 ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ a + b b + c c + a