HMMT 二月 2014 · 冲刺赛 · 第 6 题

英文解析

6. [ 5 ] Find all integers n for which is an integer.
   n − 42
   3 22
   ( n +2)( n − 2 n +4)
   n +8 n − 2 n +4
   Answer: 0, 1, 3, 4, 6 We have = = for all n 6 = − 2. Thenn − 4 ( n +2)( n − 2) n − 22
   n − 2 n +4 4 42 = n + , which is an integer if and only if is an integer. This happens whenn − 2 n − 2 n − 2
   n − 2 = − 4 , − 2 , − 1 , 1 , 2 , 4, corresponding to n = − 2 , 0 , 1 , 3 , 4 , 6, but we have n 6 = − 2 so the answersare 0 , 1 , 3 , 4 , 6.