HMMT 二月 2013 · 代数 · 第 6 题
HMMT February 2013 — Algebra — Problem 6
题目详情
- Find the number of integers n such that ⌊ ⌋ ⌈ ⌉ 100 n 99 n 1 + = . 101 100
解析
- Find the number of integers n such that ⌊ ⌋ ⌈ ⌉ 100 n 99 n 1 + = . 101 100 99 n 100 n 99 n Answer: 10100 Consider f ( n ) = ⌈ ⌉ − ⌊ ⌋ . Note that f ( n + 10100) = ⌈ + 99 · 101 ⌉ − 100 101 100 100 n 2 2 ⌊ + 100 ⌋ = f ( n ) + 99 · 101 − 100 = f ( n ) − 1. Thus, for each residue class r modulo 10100, there 101 is exactly one value of n for which f ( n ) = 1 and n ≡ r (mod 10100). It follows immediately that the answer is 10100.