International Mathematical Olympiad 2002 Problem 4

The positive divisors of the integer $n>1$ are $d_{1}<d_{2}<\ldots<d_{k}$, so that $d_{1}=1,d_{k}=n$. Let $d=d_{1}d_{2}+d_{2}d_{3}+\cdots+d_{k-1}d_{k}$. Show that $d<n^{2}$ and find all $n$ for which $d$ divides $n^{2}$.