International Mathematical Olympiad 2004 Problem 4

Let $n \geq 3$ be an integer. Let $t_1, t_2, \ldots, t_n$ be positive real numbers such that $$n^2 + 1 > (t_1 + t_2 + \ldots + t_n)\left(\frac{1}{t_1} + \frac{1}{t_2} + \ldots + \frac{1}{t_n}\right).$$

Show that $t_i, t_j, t_k$ are side lengths of a triangle for all $i$, $j$, $k$ with $1 \leq i < j < k \leq n$.