Question 82
If the real numbers $x_{1}, x_{2}, \ldots, x_{2008}$ are such that $0 \lt x_{i} \lt 1$, for any $i$, show that
$$
1 + \sum_{1 \leq i \lt j \leq 2008} x_{i} x_{j} \gt \sum_{i=1}^{2008} x_{i}
$$
$$
1 + \sum_{1 \leq i \lt j \leq 2008} x_{i} x_{j} \gt \sum_{i=1}^{2008} x_{i}
$$