HMMT 二月 2006 · 代数 · 第 2 题

英文解析

2. Find all real solutions ( x, y ) of the system x + y = 12 = y + x.
   ( ) ( )
   √ √ √ √
   1 + 3 5 1 − 3 5 1 − 3 5 1 + 3 5
   Answer: (3 , 3), ( − 4 , − 4), , , ,
   2 2 2 2
   2 2
   Solution: We have x + y = y + x which can be written as ( x − y )( x + y − 1) = 0. Thecase x = y yields x + x − 12 = 0, hence ( x, y ) = (3 , 3) or ( − 4 , − 4). The case y = 1 − x 2
   √ √
   1 ± 1+44
   1 ± 3 5
   2 2
   yields x + 1 − x − 12 = x − x − 11 = 0 which has solutions x = = .
   2 2
   The other two solutions follow.