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== Problem ==
The [[equation]] <math> 2^{333x-2} + 2^{111x+2} = 2^{222x+1} + 1 </math> has three [[real]] [[root]]s. Given that their sum is <math> \frac mn </math> where <math> m </math> and <math> n </math> are [[relatively prime]] [[positive integer]]s, find <math> m+n. </math>

== Solution ==
Let <math>y = 2^{111x}</math>.  Then our equation reads <math>\frac{1}{4}y^3 + 4y = 2y^2 + 1</math> or <math>y^3 - 8y^2 + 16y - 4 = 0</math>.  Thus, if this equation has roots <math>r_1, r_2</math> and <math>r_3</math>, by [[Vieta's formulas]] we have <math>r_1\cdot r_2\cdot r_3 = 4</math>.  Let the corresponding values of <math>x</math> be <math>x_1, x_2</math> and <math>x_3</math>.  Then the previous statement says that <math>2^{111\cdot(x_1 + x_2 + x_3)} = 4</math> so that taking a [[logarithm]] gives <math>111(x_1 + x_2 + x_3) = 2</math> and <math>x_1 + x_2 + x_3 = \frac{2}{111}</math>.  Thus the answer is <math>111 + 2 = 113</math>.

== See also ==
{{AIME box|year=2005|n=I|num-b=7|num-a=9}}

[[Category:Intermediate Algebra Problems]]