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Turkey NMO 2008 1st Round - P22 (Number Theory)

Source:

August 26, 2012

Problem Statement

How many pairs of postive integers (a,b)(a,b) with aba\geq b are there such that a2+b2a^2+b^2 divides both a3+ba^3+b and a+b3a+b^3?
<spanclass=latexbold>(A)</span> 0<spanclass=latexbold>(B)</span> 1<spanclass=latexbold>(C)</span> 2<spanclass=latexbold>(D)</span> 3<spanclass=latexbold>(E)</span> Infinitely many <span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ 3 \qquad<span class='latex-bold'>(E)</span>\ \text{Infinitely many}