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HMMT 二月 2008 · 代数 · 第 2 题

HMMT February 2008 — Algebra — Problem 2

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 3 ] Let f ( n ) be the number of times you have to hit the key on a calculator to get a number lessthan 2 starting from n . For instance, f (2) = 1 , f (5) = 2. For how many 1 < m < 2008 is f ( m ) odd? 2
解析

英文解析

  1. [ 3 ] Let f ( n ) be the number of times you have to hit the key on a calculator to get a number lessthan 2 starting from n . For instance, f (2) = 1 , f (5) = 2. For how many 1 < m < 2008 is f ( m ) odd?
    1 2 4 8 16 32 8 16
    Answer: 242 This is [2 , 2 ) ∪ [2 , 2 ) ∪ [2 , 2 ) . . . , and 2 < 2008 < 2 so we have exactly the first two intervals. 2