IMO 2004 Problem 4

Let $n \geq 3$ be an integer. Let $t_1$ , $t_2$ , ..., $t_n$ be positive real numbers such that $$n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right).$$ Show that $t_i$ , $t_j$ , $t_k$ are side lengths of a triangle for all $i$ , $j$ , $k$ with $1 \leq i < j < k \leq n$ .

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