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a^2+b^2+c^2+d^2 = 1989

Source: IMO Shortlist 1989, Problem 15, ILL 50

September 18, 2008

number theorysystem of equationsDiophantine equationAdditive Number TheoryIMO Shortlist

Problem Statement

Let a, b, c, d,m, n \in \mathbb{Z}^\plus{} such that a^2\plus{}b^2\plus{}c^2\plus{}d^2 \equal{} 1989, a\plus{}b\plus{}c\plus{}d \equal{} m^2, and the largest of

a, b, c, d

n^2.

Determine, with proof, the values of

m

and

n.

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