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2012 AMC 10AProblem 24P21-25~8 minPrint
Algebraic ManipulationDiophantine EquationsSystems of EquationsApply an identity: SFFT, sum of squares, difference of cubes, VietaSplit into exhaustive cases and handle eachSubstitute to simplify (u = x+1/x, shifting, scaling)

Let aa , bb , and cc be positive integers with a≥a\ge b≥b\ge cc such that \begin{align}a^2-b^2-c^2+ab&=2011\text{ and}\\ a^2+3b^2+3c^2-3ab-2ac-2bc&=-1997.\end{align}

What is aa ?

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Problem © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. Solution and commentary are original to this site.