HMMT 十一月 2013 · 冲刺赛 · 第 17 题
HMMT November 2013 — Guts Round — Problem 17
题目详情
- [ 10 ] Find the sum of over all positive integers n with the property that the decimal representation n 1 of terminates. n
解析
- [ 10 ] Find the sum of over all positive integers n with the property that the decimal representation n 1 of terminates. n 5 1 i j Answer: The decimal representation of terminates if and only if n = 2 5 for some nonnegative 2 n integers i, j , so our desired sum is ∑ ∑ ∑ ∑ 2 5 5 − i − j − i − j − 1 − 1 − 1 − 1 2 5 = 2 5 = (1 − 2 ) (1 − 5 ) = = . 1 4 2 i ≥ 0 j ≥ 0 i ≥ 0 j ≥ 0