Question 61
Given an integer $n \geq 2$, determine the number of ordered $n$-tuples of integers $(a_{1}, a_{2}, \ldots, a_{n})$ such that
a) $a_{1} + a_{2} + \cdots + a_{n} \geq n^{2}$; and
b) $a_{1}^{2} + a_{2}^{2} + \cdots + a_{n}^{2} \leq n^{3} + 1$.
a) $a_{1} + a_{2} + \cdots + a_{n} \geq n^{2}$; and
b) $a_{1}^{2} + a_{2}^{2} + \cdots + a_{n}^{2} \leq n^{3} + 1$.