BALTIC-WAY 2007 Problem 1

For a positive integer $n$ consider any partition of the set $\{ 1,2,\ldots ,2n \}$ into $n$ two-element subsets $P_1,P_2\ldots,P_n$ . In each subset $P_i$ , let $p_i$ be the product of the two numbers in $P_i$ . Prove that $$\frac{1}{p_1}+\frac{1}{p_2}+\ldots + \frac{1}{p_n}<1$$