Putnam 1966

Let $a, b, c$ be the lengths of the sides of a triangle, let $p = \frac{a+b+c}{2}$, and $r$ be the radius of the inscribed circle. Find the upper bound of [ \frac{1}{(p-a)^2} + \frac{1}{(p-b)^2} + \frac{1}{(p-c)^2} ] in terms of the radius.

Let a convex polygon $P$ be contained in a square of side 1. Find the upper bound of the sum of the squares of the sides of $P$.