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Mock AIME 5 Pre 2005 Problems/Problem 8

Revision as of 00:02, 15 February 2024 by Abbywong (talk | contribs) (Created page with "Square both sides to give <math>x^2+2x\sqrt{y}+y=22+2\sqrt{96}</math>. Thus, we have the system of equations <math>x^2+y=22</math> <math>x^2y=96</math>. Thus, we have <math>y(...")
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Square both sides to give $x^2+2x\sqrt{y}+y=22+2\sqrt{96}$. Thus, we have the system of equations $x^2+y=22$ $x^2y=96$. Thus, we have $y(22-y)=96$, $y^2-22y+96=0$, $(y-6)(y-16)=0$. Thus, we have $y=6$ as $x$ is an integer, and so $x=4$. Thus, we have $\frac{x}{y}=\frac{4}{6}=\frac{2}{3}$, so the answer is $2+3=\boxed{5}$. ~AbbyWong

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