PUMaC 2012 · 代数（B 组） · 第 5 题

PUMaC 2012 — Algebra (Division B) — Problem 5

[ 5 ] Considering all numbers of the form n = b c , where b x c denotes the greatest integer less 2012 than or equal to x , and k ranges from 1 to 2012, how many of these n ’s are distinct?

解析

[ 5 ] Considering all numbers of the form n = b c , where b x c denotes the greatest integer less 2012 than or equal to x , and k ranges from 1 to 2012, how many of these n ’s are distinct? Solution: 3 3 3 ( k + 1) k 3 k + 3 k + 1 − = 2012 2012 2012 3 3 k + 3 k + 1 2 ≥ 1 ⇐⇒ 3 k + 3 k + 1 ≥ 2012 2012 2 2 ⇐⇒ 3 k + 3 k ≥ 2011 ⇐⇒ 3 k + 3 k ≥ 2010 2 ⇐⇒ k + k ≥ 670 ⇐⇒ k ≥ 26 Thus, for k of at least 26, the difference between two consecutive fractions is at least 1, so the difference between their integer parts is also at least 1, so the numbers are different; in conclusion, for k between 26 and 2012, there are 1987 different numbers. For k less than 26, the difference between two consecutive fractions is less than 1, so the integer 3 3 k k parts of two consecutive fractions is at most 1. For k = 1, [ ] = 0, and for k = 25, [ ] = 7, 2012 2012 so for k less than 26, there are 8 different numbers. In the end, we get that for k ranging from 1 to 2012, there are 1995 different numbers in the sequence. Answer: 1995