HMMT 十一月 2010 · GEN1 赛 · 第 6 题

英文解析

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 lessthan 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 ofb 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 isgreater 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 2
   √
   √
   1+ 41
   solution x = . Since 3 < 41 < 7, this number is between 1 and 2, so it satisfies the equation.
   √4
   1+ 11
   If b x c = 2, then our equation becomes 2 x − 2 x = 5. We find the positive solution x = . Since 2
   √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