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an^3 + bn^2 + cn + d is integer, a is rational

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April 23, 2009
algebrapolynomial

Problem Statement

Let a,b,c,d a,b,c,d be rational numbers with a>0 a>0. If for every integer n≥0 n\ge 0, the number an^{3} \plus{}bn^{2} \plus{}cn\plus{}d is also integer, then the minimal value of a a will be
<spanclass=′latex−bold′>(A)</span> 1<spanclass=′latex−bold′>(B)</span> 12<spanclass=′latex−bold′>(C)</span> 16<spanclass=′latex−bold′>(D)</span> Cannot be found<spanclass=′latex−bold′>(E)</span> None<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ \frac{1}{2} \qquad<span class='latex-bold'>(C)</span>\ \frac{1}{6} \qquad<span class='latex-bold'>(D)</span>\ \text{Cannot be found} \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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