Problem

Source: IMO Shortlist 1989, Problem 15, ILL 50

Tags: number theory, system of equations, Diophantine equation, Additive Number Theory, IMO Shortlist



Let $ a, b, c, d,m, n \in \mathbb{Z}^+$ such that \[ a^2+b^2+c^2+d^2 = 1989,\] \[ a+b+c+d = m^2,\] and the largest of $ a, b, c, d$ is $ n^2.$ Determine, with proof, the values of $m$ and $ n.$