Putnam 1978 B6

Let $p$ and $n$ be positive integers. Suppose that the numbers $c_{hk}$ ( $h=1,2,\ldots,n$ ; $k=1,2,\ldots,ph$ ) satisfy $0 \leq c_{hk} \leq 1.$ Prove that $$\left( \sum \frac{ c_{hk} }{h} \right)^2 \leq 2p \sum c_{hk} ,$$ where each summation is over all admissible ordered pairs $(h,k).$