HMMT 二月 2013 · 代数 · 第 6 题

英文解析

6. 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, thereis exactly one value of n for which f ( n ) = 1 and n ≡ r (mod 10100). It follows immediately that the 101
   answer is 10100.