BALKAN-SL 2008 A5

Consider an integer $n \geq 1$ , $a_1,a_2, \ldots , a_n$ real numbers in $[-1,1]$ satisfying \begin{align*}a_1+a_2+\ldots +a_n=0 \end{align*} and a function $f: [-1,1] \mapsto \mathbb{R}$ such \begin{align*} \mid f(x)-f(y) \mid \le \mid x-y \mid \end{align*} for every $x,y \in [-1,1]$ . Prove \begin{align*} \left| f(x) - \frac{f(a_1) +f(a_2) + \ldots + f(a_n)}{n} \right| \le 1 \end{align*} for every $x$ $\in [-1,1]$ . For a given sequence $a_1,a_2, \ldots ,a_n$ , Find $f$ and $x$ so hat the equality holds.