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 isban 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
英文原题
Show that if j is a juicy integer, then 2 j is juicy as well.
解析
英文解析
- Say that is a positive rational number in simplest form, with a 6 = 1. Further, say that n isban 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) − ba 1
Solution: − = . Therefore, when we write it in simplest form, its numeratorb 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 asban − b < 0 ⇐⇒ an < b ⇐⇒ > n , which is given.
1 a