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P27 [Algebra] - Turkish NMO 1st Round - 2013

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April 20, 2013
algebrapolynomialinequalitiestrigonometry

Problem Statement

For how many pairs (a,b)(a,b) from (1,2)(1,2), (3,5)(3,5), (5,7)(5,7), (7,11)(7,11), the polynomial P(x)=x5+ax4+bx3+bx2+ax+1P(x)=x^5+ax^4+bx^3+bx^2+ax+1 has exactly one real root?
<spanclass=′latex−bold′>(A)</span> 4<spanclass=′latex−bold′>(B)</span> 3<spanclass=′latex−bold′>(C)</span> 2<spanclass=′latex−bold′>(D)</span> 1<spanclass=′latex−bold′>(E)</span> 0 <span class='latex-bold'>(A)</span>\ 4 \qquad<span class='latex-bold'>(B)</span>\ 3 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ 1 \qquad<span class='latex-bold'>(E)</span>\ 0
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