BG-MO 1975 Problem 4

In the plane are given a circle $k$ with radii $R$ and the points $A_1,A_2,\ldots,A_n$ , lying on $k$ or outside $k$ . Prove that there exist infinitely many points $X$ from the given circumference for which $$\sum_{i=1}^n A_iX^2\ge2nR^2.$$ Does there exist a pair of points on different sides of some diameter, $X$ and $Y$ from $k$ , such that $$\sum_{i=1}^n A_iX^2\ge2nR^2\text{ and }\sum_{i=1}^n A_iY^2\ge2nR^2?$$ H. Lesov