Question 6
Let $x_1 \le x_2 \le \cdots \le x_{100}$ be real numbers such that $|x_1| + |x_2| + \cdots + |x_{100}| = 1$ and $x_1 + x_2 + \cdots + x_{100} = 0.$ Among all such $100$-tuples of numbers, the greatest value that $x_{76} - x_{16}$ can achieve is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$