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HMMT 十一月 2019 · 冲刺赛 · 第 11 题

HMMT November 2019 — Guts Round — Problem 11

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [8] Let a, b, c, d be real numbers such that min(20 x + 19 , 19 x + 20) = ( ax + b ) − | cx + d | for all real numbers x . Find ab + cd .
解析
  1. [8] Let a, b, c, d be real numbers such that min(20 x + 19 , 19 x + 20) = ( ax + b ) − | cx + d | for all real numbers x . Find ab + cd . Proposed by: Yuan Yao Answer: 380 ∣ ∣ p + q p − q ∣ ∣ In general, min( p, q ) = − . Letting p = 20 x + 19 and q = 19 x + 20 gives a = b = 19 . 5 and 2 2 2 2 c = d = ± 0 . 5. Then the answer is 19 . 5 − 0 . 5 = 19 · 20 = 380.