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Part of 2017 OMMock - Mexico National Olympiad Mock Exam

Problems(1)

Proving that system has solutions assuming xy=z^2+2

Source: Mexico National Olympiad Mock Exam 2017 P3

11/2/2019
Let x,y,zx, y, z be positive integers such that xy=z2+2xy=z^2+2. Prove that there exist integers a,b,c,da, b, c, d such that the following equalities are satisfied: \begin{eqnarray*} x=a^2+2b^2\\ y=c^2+d^2\\ z=ac+2bd\\ \end{eqnarray*}
Proposed by Isaac Jiménez
number theorySquaresSystem