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P27 [Algebra] - Turkish NMO 1st Round - 2005

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November 2, 2013

Problem Statement

What is the maximum value of the difference between the largest real root and the smallest real root of the equation system ax2+bx+c=0bx2+cx+a=0cx2+ax+b=0\begin{array}{rcl} ax^2 + bx+ c &=& 0 \\ bx^2 + cx+ a &=& 0 \\ cx^2 + ax+ b &=& 0 \end{array}, where at least one of the reals a,b,ca,b,c is non-zero?
<spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 1<spanclass=′latex−bold′>(C)</span> 2<spanclass=′latex−bold′>(D)</span> 32<spanclass=′latex−bold′>(E)</span> There is no upper-bound <span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ \sqrt 2 \qquad<span class='latex-bold'>(D)</span>\ 3\sqrt 2 \qquad<span class='latex-bold'>(E)</span>\ \text{There is no upper-bound}
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