MathNet 2000 — Math Olympiad Vault Question 68

Let $a_1 = 1$, $a_{n+1} = \frac{a_n}{n} + \frac{n}{a_n}$ for $n = 1, 2, 3, \dots$. Find the greatest integer less than or equal to $a_{2000}$. Be sure to prove your claim.