HMMT 二月 2011 · 冲刺赛 · 第 7 题

英文解析

7. [ 6 ] For any positive real numbers a and b , define a ° b = a + b + 2 ab . Find all positive real numbersx such that x ° 9 x = 121.2
   √
   31 − 3 53
   Answer:
   √2
   √ √
   2 2 2
   Since a ° b = ( a + b ) , we have x ° 9 x = ( x + 3 x ) . Moreover, since x is positive, we have
   √ √
   √ √
   − 3+ 53 31 − 3 53
   x + 3 x = 11, and the only possible solution is that x = , so x = .
   2 2