Answer: \(\lfloor\frac{n}{2}\rfloor^2\) if \(n\) is odd and \(\lfloor\frac{n}{2}\rfloor^2 - \lfloor\frac{n}{2}\rfloor\) otherwise. Estimation. Notice that \(a_i\) can be the midpoint of at most \(\min(i-1, n-i)\) pairs \((a_j, a_k)\) with \(j<i<k\). Hence balanced pairs may not be more than \[\sum_{i=1}^n\min(i-1, n-i)=\begin{cases} \lfloor\frac{n}{2}\rfloor^2, &\text{ if } n \text{ is odd}, \\ \lfloor\frac{n}{2}\rfloor^2 - \lfloor\frac{n}{2}\rfloor, &\text{ otherwise}. \end{cases}\]Construction. Take any equally spaced sequence, for example \(a_i=i\) for all \(i\).