HMMT 二月 2016 · 冲刺赛 · 第 30 题

HMMT February 2016 — Guts Round — Problem 30

[ 16 ] Determine the number of triples 0 ≤ k, m, n ≤ 100 of integers such that m n k 2 n − 2 m = 2 .

解析

[ 16 ] Determine the number of triples 0 ≤ k, m, n ≤ 100 of integers such that m n k 2 n − 2 m = 2 . Proposed by: Casey Fu Answer: 22 m d First consider when n ≥ m , so let n = m + d where d ≥ 0 . Then we have 2 ( m + d − 2 m ) = m d j 2 ( m (1 − 2 ) + d ), which is non-positive unless m = 0. So our first set of solutions is m = 0 , n = 2 . m n n + d Now, we can assume that m > n , so let m = n + d where d > 0. Rewrite 2 n − 2 m = 2 n − n n d d 2 ( n + d ) = 2 ((2 − 1) n − d ). In order for this to be a power of 2, (2 − 1) n − d must be a power of j d d