HMMT 2014 A7

Find the largest real number $c$ such that

$$\sum_{i=1}^{101} x_{i}^{2} \geq c M^{2}$$

whenever $x_{1}, \ldots, x_{101}$ are real numbers such that $x_{1}+\cdots+x_{101}=0$ and $M$ is the median of $x_{1}, \ldots, x_{101}$.