Putnam 1977 B5

Problem B-5 Suppose that $a_{1}, a_{2}, \ldots, a_{n}$ are real $(n>1)$ and $$A+\sum_{i=1}^{n} a_{i}^{2}<\frac{1}{n-1}\left(\sum_{i=1}^{n} a_{i}\right)^{2}$$

Prove that $A<2 a_{i} a_{j}$ for $1<i<j<n$.