International Mathematical Olympiad 2002 — Official contest Question 4

Let $n \ge 2$ be a positive integer with divisors $1 = d_1 \lt d_2 \lt \dots \lt d_k = n$. Prove that $d_1d_2 + d_2d_3 + \dots + d_{k-1} d_k$ is always less than $n^2$, and determine when it is a divisor of $n^2$.