HMMT 十一月 2017 · 冲刺赛 · 第 11 题

HMMT November 2017 — Guts Round — Problem 11

[ 4 ] Find the sum of all real numbers x for which 1 bb· · · bbb x c + x c + x c · · · c + x c = 2017 and {{· · · {{{ x } + x } + x } · · · } + x } = 2017 where there are 2017 x ’s in both equations. ( b x c is the integer part of x , and { x } is the fractional part of x .) Express your sum as a mixed number.

解析

[ 4 ] Find the sum of all real numbers x for which 1 bb· · · bbb x c + x c + x c · · · c + x c = 2017 and {{· · · {{{ x } + x } + x } · · · } + x } = 2017 where there are 2017 x ’s in both equations. ( b x c is the integer part of x , and { x } is the fractional part of x .) Express your sum as a mixed number. Proposed by: Yuan Yao 1 6101426 Answer: 3025 or 2017 2017 1 The two equations are equivalent to 2017 b x c = 2017 and { 2017 x } = , respectively. The first 2017 equation reduces to b x c = 1, so we must have x = 1 + r for some real r satisfying 0 ≤ r < 1. From the 1 1 second equation, we deduce that { 2017 x } = { 2017 + 2017 r } = { 2017 r } = , so 2017 r = n + , 2017 2017 n 1 where n is an integer. Dividing both sides of this equation by 2017 yields r = + , where 2 2017 2017 n 1 n = 0 , 1 , 2 , . . . , 2016 so that we have 0 ≤ r < 1. Thus, we have x = 1 + r = 1 + + 2 2017 2017 2016 · 2017 1 1 for n = 0 , 1 , 2 , . . . , 2016. The sum of these solutions is 2017 · 1 + · + 2017 · = 2 2 2017 2017 1 2016 1 2017 + + = 3025 . 2 2017 2017