HMMT 十一月 2015 · 团队赛 · 第 3 题
HMMT November 2015 — Team Round — Problem 3
题目详情
英文原题
- [ 3 ] Let b x c denote the largest integer less than or equal to x , and let { x } denote the fractional part ofx . For example, b π c = 3, and { π } = 0 . 14159 . . . , while b 100 c = 100 and { 100 } = 0. If n is the largestb n c
2015
solution to the equation = , compute { n } .
n 2016
解析
英文解析
- [ 3 ] Let b x c denote the largest integer less than or equal to x , and let { x } denote the fractional part ofx . For example, b π c = 3, and { π } = 0 . 14159 . . . , while b 100 c = 100 and { 100 } = 0. If n is the largestb n c
2015
solution to the equation = , compute { n } .
n 2016
Proposed by: Alexander Katz
2014
Answer:
2015
Note that n = b n c + { n } , sob n c b n cn b n c + { n }=
2015
2016=
= ⇒ 2016 b n c = 2015 b n c + 2015 { n } = ⇒ b n c = 2015 { n }
2016
Hence, n = b n c + { n } = b n c , and so n is maximized when b n c is also maximized. As b n c is an
2015
2014
b n cinteger, and { n } < 1, the maximum possible value of b n c is 2014. Therefore, { n } = = .
2015
2015