Difference between revisions of "2018 AIME I Problems/Problem 5"
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For each ordered pair of real numbers <math>(x,y)</math> satisfying <cmath>\log_2(2x+y) = \log_4(x^2+xy+7y^2)</cmath>there is a real number <math>K</math> such that <cmath>\log_3(3x+y) = \log_9(3x^2+4xy+Ky^2).</cmath>Find the product of all possible values of <math>K</math>. | For each ordered pair of real numbers <math>(x,y)</math> satisfying <cmath>\log_2(2x+y) = \log_4(x^2+xy+7y^2)</cmath>there is a real number <math>K</math> such that <cmath>\log_3(3x+y) = \log_9(3x^2+4xy+Ky^2).</cmath>Find the product of all possible values of <math>K</math>. | ||
− | ==Solution== | + | ==Solution 1== |
− | + | Using the logarithmic property <math>\log_{a^n}b^n = \log_{a}b</math>, we note that <cmath>(2x+y)^2 = x^2+xy+7y^2</cmath> | |
− | + | That gives <cmath>x^2+xy-2y^2=0</cmath> upon simplification and division by <math>3</math>. Factoring <math>x^2+xy-2y^2=0</math> gives <cmath>(x+2y)(x-y)=0</cmath> Then, <cmath>x=y \text{ or }x=-2y</cmath> | |
− | That gives < | + | From the second equation, <cmath>9x^2+6xy+y^2=3x^2+4xy+Ky^2</cmath> If we take <math>x=y</math>, we see that <math>K=9</math>. If we take <math>x=-2y</math>, we see that <math>K=21</math>. The product is <math>\boxed{189}</math>. |
− | From the second equation, < | ||
-expiLnCalc | -expiLnCalc | ||
− | == | + | ==Solution 2== |
− | + | Do as done in Solution 1 to get <cmath>x^2+xy-2y^2=0</cmath> <cmath>\implies (\frac{x}{y})^2+\frac{x}{y}-2=0</cmath> <cmath>\implies \frac{x}{y}=\frac{-1\pm\sqrt{1+8}}{2}=1,-2</cmath> Do as done in Solution 1 to get <cmath>9x^2+6xy+y^2=3x^2+4xy+Ky^2</cmath> <cmath>\implies 6x^2+2xy+(1-K)y^2=0</cmath> <cmath>\implies 6(\frac{x}{y})^2+2\frac{x}{y}+(1-K)=0</cmath> <cmath>\implies \frac{x}{y}=\frac{-2\pm \sqrt{4-24(1-K)}}{12}</cmath> <cmath>\implies \frac{x}{y}=\frac{-2\pm 2\sqrt{6K-5}}{12}=\frac{-1\pm \sqrt{6K-5}}{6}</cmath>If <math>\frac{x}{y}=1</math> then <cmath>1=\frac{-1\pm \sqrt{6K-5}}{6}</cmath> <cmath>\implies 6=-1\pm \sqrt{6K-5}</cmath> <cmath>\implies 7=\pm \sqrt{6K-5}</cmath> <cmath>\implies 49=6K-5</cmath> <cmath>\implies K=9</cmath>If <math>\frac{x}{y}=-2</math> then <cmath>-2=\frac{-1\pm \sqrt{6K-5}}{6}</cmath> <cmath>\implies -12=-1\pm \sqrt{6K-5}</cmath> <cmath>\implies -11=\sqrt{6K-5}</cmath> <cmath>\implies 121=6K-5</cmath> <cmath>\implies 126=6K</cmath> <cmath>\implies K=21</cmath>Hence our final answer is <math>21\cdot 9=\boxed{189}</math> | |
− | + | -vsamc<math>\newline</math> | |
+ | ==Solution 3 (Official MAA)== | ||
+ | Because <math>x^2+xy+7y^2=\left(x+\tfrac{y}{2}\right)^2+\tfrac{27}{4}y^2>0,</math> the right side of the first equation is real. It follows that the left side of the equation is also real, so <math>2x+y>0</math> and <cmath>\log_2(2x+y)=\log_{2^2}(2x+y)^2=\log_4(4x^2+4xy+y^2).</cmath> Thus <math>4x^2+4xy+y^2=x^2+xy+7y^2,</math> which implies that <math>0=x^2+xy-2y^2=(x+2y)(x-y).</math> Therefore either <math>x=-2y</math> or <math>x=y,</math> and because <math>2x+y>0,</math> <math>x</math> must be positive and <math>3x+y=x+(2x+y)>0.</math> Similarly, <cmath>\log_3(3x+y)=\log_{3^2}(3x+y)^2=\log_9(9x^2+6xy+y^2).</cmath> If <math>x=-2y\ne 0,</math> then <math>9x^2+6xy+y^2=36y^2-12y^2+y^2=25y^2=3x^2+4xy+Ky^2</math> when <math>K=21.</math> If <math>x=y\ne 0,</math> then <math>9x^2+6xy+y^2=16y^2=3x^2+4xy+Ky^2</math> when <math>K=9.</math> The requested product is <math>21\cdot9=189.</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}} |
Latest revision as of 12:58, 14 August 2024
Problem 5
For each ordered pair of real numbers satisfying there is a real number such that Find the product of all possible values of .
Solution 1
Using the logarithmic property , we note that That gives upon simplification and division by . Factoring gives Then, From the second equation, If we take , we see that . If we take , we see that . The product is .
-expiLnCalc
Solution 2
Do as done in Solution 1 to get Do as done in Solution 1 to get If then If then Hence our final answer is -vsamc
Solution 3 (Official MAA)
Because the right side of the first equation is real. It follows that the left side of the equation is also real, so and Thus which implies that Therefore either or and because must be positive and Similarly, If then when If then when The requested product is
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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