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Three Positive Integers, Two Equations

Source: 2012 AMC10A Problem #24

February 8, 2012
AMCAMC 12AMC 12 AAMC 10

Problem Statement

Let a,b,a,b, and cc be positive integers with a≥b≥ca\ge b\ge c 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?
<spanclass=′latex−bold′>(A)</span> 249<spanclass=′latex−bold′>(B)</span> 250<spanclass=′latex−bold′>(C)</span> 251<spanclass=′latex−bold′>(D)</span> 252<spanclass=′latex−bold′>(E)</span> 253 <span class='latex-bold'>(A)</span>\ 249 \qquad<span class='latex-bold'>(B)</span>\ 250 \qquad<span class='latex-bold'>(C)</span>\ 251 \qquad<span class='latex-bold'>(D)</span>\ 252 \qquad<span class='latex-bold'>(E)</span>\ 253
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