HMMT 十一月 2008 · 团队赛 · 第 4 题
HMMT November 2008 — Team Round — Problem 4
题目详情
- Say that is a positive rational number in simplest form, with a 6 = 1. Further, say that n is b an integer such that: 1 a 1
n b n + 1 1 a Show that when − is written in simplest form, its numerator is smaller than a . b n +1
解析
- Say that is a positive rational number in simplest form, with a 6 = 1. Further, say that n is b an integer such that: 1 a 1
n b n + 1 a 1 Show that when − is written in simplest form, its numerator is smaller than a . b n +1 a ( n +1) − b a 1 Solution: − = . Therefore, when we write it in simplest form, its numerator b n +1 b ( n +1) will be at most a ( n + 1) − b . We claim that a ( n + 1) − b < a . Indeed, this is the same as b an − b < 0 ⇐⇒ an < b ⇐⇒ > n , which is given. a 1