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HMMT 十一月 2022 · 冲刺赛 · 第 5 题

HMMT November 2022 — Guts Round — Problem 5

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [6] Suppose x and y are positive real numbers such that 1 2 x + = y + = 3 . y x Compute the maximum possible value of xy .
解析
  1. [6] Suppose x and y are positive real numbers such that 1 2 x + = y + = 3 . y x Compute the maximum possible value of xy . Proposed by: Rishabh Das √ Answer: 3 + 7 C +2 Solution 1: Rewrite the equations as xy + 1 = 3 y and xy + 2 = 3 x . Let xy = C , so x = and 3 C +1 y = . Then 3 C + 2 C + 1 2 = C = ⇒ C − 6 C + 2 = 0 . 3 3 √ The larger of its two roots is 3 + 7. 2 2 Solution 2: Multiply the two equations to get xy +3+ = 9, so letting C = xy gives C − 6 C +2 = 0 , xy √ which has larger root C = 3 + 7.