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a_1 + a_2 + .. + a_n >= n^2 and a_1^2 + a_2^2 + ... + a_n^2 <= n^3 + 1

Source: 2013 Saudi Arabia IMO TST III p2

July 23, 2020
inequalitiesalgebracombinatorics

Problem Statement

Given an integer n≥2n \ge 2, determine the number of ordered nn-tuples of integers (a1,a2,...,an)(a_1, a_2,...,a_n) such that (a) a1+a2+..+an≥n2a_1 + a_2 + .. + a_n \ge n^2 and (b) a12+a22+...+an2≤n3+1a_1^2 + a_2^2 + ... + a_n^2 \le n^3 + 1
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