HMMT 十一月 2015 · 团队赛 · 第 3 题

HMMT November 2015 — Team Round — Problem 3

[ 3 ] Let b x c denote the largest integer less than or equal to x , and let { x } denote the fractional part of x . For example, b π c = 3, and { π } = 0 . 14159 . . . , while b 100 c = 100 and { 100 } = 0. If n is the largest b n c 2015 solution to the equation = , compute { n } . n 2016

解析

[ 3 ] Let b x c denote the largest integer less than or equal to x , and let { x } denote the fractional part of x . For example, b π c = 3, and { π } = 0 . 14159 . . . , while b 100 c = 100 and { 100 } = 0. If n is the largest b 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 } , so b n c b n c = n 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 c integer, and { n } < 1, the maximum possible value of b n c is 2014. Therefore, { n } = = . 2015 2015