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Turkey NMO 2000 1st Round - P22 (Number Theory)

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July 25, 2012

Problem Statement

How many ordered positive integer triples (x,y,z)(x,y,z) are there such that 3x2−2y2−4z2+54=05x2−3y2−7z2+74=0 \begin{array}{rcl} 3x^2-2y^2-4z^2+54 = 0 \\ 5x^2-3y^2-7z^2+74 = 0 \end{array}
<spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 3<spanclass=′latex−bold′>(D)</span> Infinitely Many<spanclass=′latex−bold′>(E)</span> None of the options <span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ \text{Infinitely Many} \qquad<span class='latex-bold'>(E)</span>\ \text{None of the options}
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