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Difference between revisions of "2018 AIME I Problems/Problem 5"

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==Note==
 
==Note==
  
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The cases <math>x=y</math> and <math>x=-2y</math> can be found by SFFT ([[https://artofproblemsolving.com/wiki/index.php/Simon%27s_Favorite_Factoring_Trick Simon's Favorite Factoring Trick]]) from <math>x^2+xy-2y^2=0 \implies (x+2y)(x-y)=0</math>.
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The cases <math>x=y</math> and <math>x=-2y</math> can be found by SFFT ([https://artofproblemsolving.com/wiki/index.php/Simon%27s_Favorite_Factoring_Trick Simon's Favorite Factoring Trick]) from <math>x^2+xy-2y^2=0 \implies (x+2y)(x-y)=0</math>.
  
 
==See Also==
 
==See Also==
 
{{AIME box|year=2018|n=I|num-b=4|num-a=6}}
 
{{AIME box|year=2018|n=I|num-b=4|num-a=6}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 11:23, 5 July 2018

For each ordered pair of real numbers $(x,y)$ satisfying \[\log_2(2x+y) = \log_4(x^2+xy+7y^2)\]there is a real number $K$ such that \[\log_3(3x+y) = \log_9(3x^2+4xy+Ky^2).\]Find the product of all possible values of $K$.

Solution

Note that $(2x+y)^2 = x^2+xy+7y^2$. That gives $x^2+xy-2y^2=0$ upon simplification and division by $3$. Then, $x=y$ or $x=-2y$. From the second equation, $9x^2+6xy+y^2=3x^2+4xy+Ky^2$. If we take $x=y$, we see that $K=9$. If we take $x=-2y$, we see that $K=21$. The product is $\boxed{189}$.

-expiLnCalc

Note

The cases $x=y$ and $x=-2y$ can be found by SFFT (Simon's Favorite Factoring Trick) from $x^2+xy-2y^2=0 \implies (x+2y)(x-y)=0$.

See Also

2018 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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