HMMT 二月 2008 · 冲刺赛 · 第 1 题

英文解析

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 a can be found by a = (10 + b ).1 2