HMMT 十一月 2010 · GEN1 赛 · 第 6 题

HMMT November 2010 — GEN1 Round — Problem 6

[ 5 ] What is the sum of the positive solutions to 2 x − x b x c = 5, where b x c is the largest integer less than or equal to x ? 2010 2

解析

[ 5 ] What is the sum of the positive solutions to 2 x − x b x c = 5, where b x c is the largest integer less than or equal to x ? √ √ 3+ 41+2 11 2 2 2 2 Answer: We first note that b x c ≤ x , so 2 x − x b x c ≥ 2 x − x = x . Since this function 4 √ is increasing on the positive reals, all solutions must be at most 5. This gives us 3 possible values of b x c : 0, 1, and 2. √ 2 5 If b x c = 0, then our equation becomes 2 x = 5, which has positive solution x = . This number is 2 greater than 1, so its floor is not 0; thus, there are no solutions in this case. 2 If b x c = 1, then our equation becomes 2 x − x = 5. Using the quadratic formula, we find the positive √ √ 1+ 41 solution x = . Since 3 < 41 < 7, this number is between 1 and 2, so it satisfies the equation. 4 √ 1+ 11 2 If b x c = 2, then our equation becomes 2 x − 2 x = 5. We find the positive solution x = . Since 2 √ 3 < 11 < 5, this number is between 2 and 3, so it satisfies the equation. √ √ √ √ 1+ 41 1+ 11 3+ 41+2 11 We then find that the sum of positive solutions is + = . 4 2 4 2010 2