Problem

Source: China Western Mathematical Olympiad 2002 P6

Tags: modular arithmetic, inequalities, number theory proposed, number theory



Given a positive integer $ n$, find all integers $ (a_{1},a_{2},\cdots,a_{n})$ satisfying the following conditions: $ (1): a_{1}+a_{2}+\cdots+a_{n}\ge n^2;$ $ (2): a_{1}^2+a_{2}^2+\cdots+a_{n}^2\le n^3+1.$