PUMaC 2011 · 代数（A 组） · 第 5 题
PUMaC 2011 — Algebra (Division A) — Problem 5
题目详情
- [ 5 ] Let 1 f ( x ) = and f ( x ) = 1 − x 1 2 x Let H be the set of all compositions of the form h ◦ h ◦ . . . ◦ h , where each h is either f or 1 2 k i 1 ( n ) f . For all h in H , let h denote h composed with itself n times. Find the greatest integer 2 ( N ) N such that π, h ( π ) , . . . , h ( π ) are all distinct for some h in H . ∞
解析
- Observe that f ( x ) = f ( x ) = x . So if h = f ◦ . . . ◦ f , then we can suppose the sequence i i 1 2 1 k i , . . . , i alternates between 1 and 2. If k is odd, then i = i , so 1 k 1 k (2) h ( x ) = ( f ◦ . . . ◦ f ◦ f ◦ . . . ◦ f )( x ) = x. i i i i 1 k 1 k ( n ) ( n ) If k is even, then either h = ( f ◦ f ) or h = ( f ◦ f ) for some n . 1 2 2 1