Question 31
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 $x_{1}, \ldots, x_{101}$.
$$
\sum_{i=1}^{101} x_{i}^{2} \geq c M^{2}
$$
whenever $x_{1}, \ldots, x_{101}$ are real numbers such that $x_{1}+\cdots+x_{101}=0$ and $M$ is the median of $x_{1}, \ldots, x_{101}$.