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HMMT 二月 2008 · 冲刺赛 · 第 1 题

HMMT February 2008 — Guts Round — Problem 1

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 5 ] Determine all pairs ( a, b ) of real numbers such that 10 , a, b, ab is an arithmetic progression.
解析
  1. [ 5 ] Determine all pairs ( a, b ) of real numbers such that 10 , a, b, ab is an arithmetic progression. 5 1 Answer: (4 , − 2) , ( , − 5) Since 10 , a, b is an arithmetic progression, we have a = (10 + b ). Also, 2 2 we have a + ab = 2 b , and so a (1 + b ) = 2 b . Substituting the expression for a gives (10 + b )(1 + b ) = 4 b . Solving this quadratic equation gives the solutions b = − 2 and b = − 5. The corresponding values for 1 a can be found by a = (10 + b ). 2