HMMT 二月 2013 · 团队赛 · 第 1 题

英文解析

1. [ 10 ] Let a and b be real numbers such that = . Find all possible values of .
   2 2 2 2
   a + b 4 a + b
   √
   Answer: The hypothesis statement is equivalent to 3
   2 22
   a + b = 4 ab
   1 : ( a + b ) = 6 ab 2
   2 : ( a − b ) = 2 ab 2
   Multiplying equations 1 and 2,
   2 2 2 2
   ( a − b ) = 12( ab )
   √
   2 2
   | a − b | = ± 12 ab
   Since the left hand side and ab are both positive,
   √
   2 2
   | a − b | = 12 ab
   √ √ √
   2 2
   | a − b | 12 ab 12 3 = = =
   2 2 2 2
   a + b a + b 4 2
   (It is clear that such a and b exist: for example, we can take a = 1 and solve for b by way of thequadratic formula.)