Problems

Let $x_1,x_2,\dots ,x_n$ be some numbers belonging to the interval $[0,1]$. Prove that on this segment there is a number $x$ such that $$\frac{1}{n}(|x-x_1|+|x-x_2|+\cdots +|x-x_n|)=1/2.$$

A convex figure and point $A$ inside it are given. Prove that there is a chord (that is, a segment joining two boundary points of a convex figure) passing through point $A$ and dividing it in half at point $A$.