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 3 years ago
AMC 10 2012 #24
Let a, b, and c be positive integers with \[a \ge b \ge c\] such that \[a^2b^2c^2+ab=2011\] and \[a^2+3b^2+3c^23ab2ac2bc=1997\]. What is a?
 3 years ago
AMC 10 2012 #24 Let a, b, and c be positive integers with \[a \ge b \ge c\] such that \[a^2b^2c^2+ab=2011\] and \[a^2+3b^2+3c^23ab2ac2bc=1997\]. What is a?

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Mr.Math
 3 years ago
Best ResponseYou've already chosen the best response.4We can write the second equation as \[(ac)^2+(ab)^2+(bc)^2+b^2+c^2a^2ab=1997.\] But \(b^2+c^2a^2ab=2011\), and that gives \((ac)^2+(ab)^2+(bc)^2=14=3^2+2^2+1^2.\) That's \(ac=3 \implies c=a3\), \(ab=2 \implies b=a2\) and \(bc=1\). Plug this into the first equation and solve the quadratic equation \[a^2(a2)^2(a3)^2+a(a2)=2011 \implies \large a=253.\]
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