HMMT 二月 2010 · 代数 · 第 1 题

英文解析

1. [ 3 ] Suppose that x and y are positive reals such that
   2 2 4
   x − y = 3 , x + y = 13 .
   Find x .
   √
   3+ 17 2 2 4 2
   Answer: Squaring both sides of x − y = 3 gives x + y − 2 xy = 9. Subtract this equation
   2 2 4 22
   from twice the second given to get x + 2 xy + y = 17 = ⇒ x + y = ± 17. Combining this equation
   √ √
   3 ± 17 3+ 17
   with the first given, we see that x = . Since x is a positive real, x must be .
   2 2
   1 1