Given an integer $n \ge 2$, determine the number of ordered $n$-tuples of integers $(a_1, a_2,...,a_n)$ such that (a) $a_1 + a_2 + .. + a_n \ge n^2$ and (b) $a_1^2 + a_2^2 + ... + a_n^2 \le n^3 + 1$
Source: 2013 Saudi Arabia IMO TST III p2
Tags: inequalities, algebra, combinatorics
Given an integer $n \ge 2$, determine the number of ordered $n$-tuples of integers $(a_1, a_2,...,a_n)$ such that (a) $a_1 + a_2 + .. + a_n \ge n^2$ and (b) $a_1^2 + a_2^2 + ... + a_n^2 \le n^3 + 1$