AIME 2018 I · 第 5 题

Solution 1

Using the logarithmic property $\log_{a^n}b^n = \log_{a}b$, we note that

$$(2x+y)^2 = x^2+xy+7y^2$$ That gives

$$x^2+xy-2y^2=0$$ upon simplification and division by $3$. Factoring $x^2+xy-2y^2=0$ gives

$$(x+2y)(x-y)=0$$ Then,

$$x=y \text{ or }x=-2y$$ 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

Solution 2

Do as done in Solution 1 to get

$$x^2+xy-2y^2=0$$ $$\implies (\frac{x}{y})^2+\frac{x}{y}-2=0$$ $$\implies \frac{x}{y}=\frac{-1\pm\sqrt{1+8}}{2}=1,-2$$ Do as done in Solution 1 to get

$$9x^2+6xy+y^2=3x^2+4xy+Ky^2$$ $$\implies 6x^2+2xy+(1-K)y^2=0$$ $$\implies 6(\frac{x}{y})^2+2\frac{x}{y}+(1-K)=0$$ $$\implies \frac{x}{y}=\frac{-2\pm \sqrt{4-24(1-K)}}{12}$$ $$\implies \frac{x}{y}=\frac{-2\pm 2\sqrt{6K-5}}{12}=\frac{-1\pm \sqrt{6K-5}}{6}$$ If $\frac{x}{y}=1$ then

$$1=\frac{-1\pm \sqrt{6K-5}}{6}$$ $$\implies 6=-1\pm \sqrt{6K-5}$$ $$\implies 7=\pm \sqrt{6K-5}$$ $$\implies 49=6K-5$$ $$\implies K=9$$ If $\frac{x}{y}=-2$ then

$$-2=\frac{-1\pm \sqrt{6K-5}}{6}$$ $$\implies -12=-1\pm \sqrt{6K-5}$$ $$\implies -11=\sqrt{6K-5}$$ $$\implies 121=6K-5$$ $$\implies 126=6K$$ $$\implies K=21$$ Hence our final answer is $21\cdot 9=\boxed{189}$ -vsamc$\newline$

Solution 3 (Official MAA)

Because $x^2+xy+7y^2=\left(x+\tfrac{y}{2}\right)^2+\tfrac{27}{4}y^2>0,$ the right side of the first equation is real. It follows that the left side of the equation is also real, so $2x+y>0$ and

$$\log_2(2x+y)=\log_{2^2}(2x+y)^2=\log_4(4x^2+4xy+y^2).$$ Thus $4x^2+4xy+y^2=x^2+xy+7y^2,$ which implies that $0=x^2+xy-2y^2=(x+2y)(x-y).$ Therefore either $x=-2y$ or $x=y,$ and because $2x+y>0,$ $x$ must be positive and $3x+y=x+(2x+y)>0.$ Similarly,

$$\log_3(3x+y)=\log_{3^2}(3x+y)^2=\log_9(9x^2+6xy+y^2).$$ If $x=-2y\ne 0,$ then $9x^2+6xy+y^2=36y^2-12y^2+y^2=25y^2=3x^2+4xy+Ky^2$ when $K=21.$ If $x=y\ne 0,$ then $9x^2+6xy+y^2=16y^2=3x^2+4xy+Ky^2$ when $K=9.$ The requested product is $21\cdot9=189.$