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Dice, Parabola

Source: 2012 AMC 12B #13

February 23, 2012
conicsparabolaprobabilitysymmetryLaTeXAMC

Problem Statement

Two parabolas have equations y=x2+ax+by=x^2+ax+b and y=x2+cx+dy=x^2+cx+d, where aa, bb, cc, and dd are integers (not necessarily different), each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas have at least one point in common?
<spanclass=latexbold>(A)</span> 12<spanclass=latexbold>(B)</span> 2536<spanclass=latexbold>(C)</span> 56<spanclass=latexbold>(D)</span> 3136<spanclass=latexbold>(E)</span> 1<span class='latex-bold'>(A)</span>\ \frac{1}{2} \qquad<span class='latex-bold'>(B)</span>\ \frac{25}{36} \qquad<span class='latex-bold'>(C)</span>\ \frac{5}{6} \qquad<span class='latex-bold'>(D)</span>\ \frac{31}{36} \qquad<span class='latex-bold'>(E)</span>\ 1