Question 81
If for the real numbers $x_{1}, x_{2}, \ldots, x_{n}$ it is $0 \lt x_{i} \lt 1$, for any $i$, show that
$$
1 + \sum_{1 \leq i \lt j \leq n} x_{i} x_{j} \gt \sum_{i=1}^{n} x_{i}
$$
$$
1 + \sum_{1 \leq i \lt j \leq n} x_{i} x_{j} \gt \sum_{i=1}^{n} x_{i}
$$