Difference between revisions of "2012 AMC 10A Problems/Problem 24"

(Created page with "== Problem 24 == Let <math>a</math>, <math>b</math>, and <math>c</math> be positive integers with <math>a\geb\gec</math> such that [\a^2-b^2-c^2+ab=2011]\ and [\a^2+3b^2+3c^2-3a...")
 
(Problem 24)
Line 2: Line 2:
  
 
Let <math>a</math>, <math>b</math>, and <math>c</math> be positive integers with <math>a\geb\gec</math> such that
 
Let <math>a</math>, <math>b</math>, and <math>c</math> be positive integers with <math>a\geb\gec</math> such that
[\a^2-b^2-c^2+ab=2011]\ and
+
<math>a^2-b^2-c^2+ab=2011</math> and
[\a^2+3b^2+3c^2-3ab-2ac-2bc=-1997]\.
+
<math>a^2+3b^2+3c^2-3ab-2ac-2bc=-1997</math>.
 +
 
 +
What is <math>a</math>?
  
 
<math> \textbf{(A)}\ 249\qquad\textbf{(B)}\ 250\qquad\textbf{(C)}\ 251\qquad\textbf{(D)}\ 252\qquad\textbf{(E)}\ 253 </math>
 
<math> \textbf{(A)}\ 249\qquad\textbf{(B)}\ 250\qquad\textbf{(C)}\ 251\qquad\textbf{(D)}\ 252\qquad\textbf{(E)}\ 253 </math>

Revision as of 20:18, 8 February 2012

Problem 24

Let $a$, $b$, and $c$ be positive integers with $a\geb\gec$ (Error compiling LaTeX. ! Undefined control sequence.) such that $a^2-b^2-c^2+ab=2011$ and $a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$.

What is $a$?

$\textbf{(A)}\ 249\qquad\textbf{(B)}\ 250\qquad\textbf{(C)}\ 251\qquad\textbf{(D)}\ 252\qquad\textbf{(E)}\ 253$

Invalid username
Login to AoPS