AIME 2010 I · 第 3 题

Solution 1

Substitute $y = \frac34x$ into $x^y = y^x$ and solve.

$$x^{\frac34x} = \left(\frac34x\right)^x$$ $$x^{\frac34x} = \left(\frac34\right)^x \cdot x^x$$ $$x^{-\frac14x} = \left(\frac34\right)^x$$ $$x^{-\frac14} = \frac34$$ $$x = \frac{256}{81}$$ $$y = \frac34x = \frac{192}{81}$$ $$x + y = \frac{448}{81}$$ $$448 + 81 = \boxed{529}$$

Solution 2

We solve in general using $c$ instead of $3/4$. Substituting $y = cx$, we have:

$$x^{cx} = (cx)^x \Longrightarrow (x^x)^c = c^x\cdot x^x$$ Dividing by $x^x$, we get $(x^x)^{c - 1} = c^x$.

Taking the $x$th root, $x^{c - 1} = c$, or $x = c^{1/(c - 1)}$.

In the case $c = \frac34$, $x = \Bigg(\frac34\Bigg)^{ - 4} = \Bigg(\frac43\Bigg)^4 = \frac {256}{81}$, $y = \frac {64}{27}$, $x + y = \frac {256 + 192}{81} = \frac {448}{81}$, yielding an answer of $448 + 81 = \boxed{529}$.

Solution 3

Taking the logarithm base $x$ of both sides, we arrive with:

$$y = \log_x y^x \Longrightarrow \frac{y}{x} = \log_{x} y = \log_x \frac{3}{4}x = \frac{3}{4}$$ Where the last two simplifications were made since $y = \frac{3}{4}x$. Then,

$$x^{\frac{3}{4}} = \frac{3}{4}x \Longrightarrow x^{\frac{1}{4}} = \frac{4}{3} \Longrightarrow x = \left(\frac{4}{3}\right)^4$$ Then, $y = \left(\frac{4}{3}\right)^3$, and thus:

$$x+y = \left(\frac{4}{3}\right)^3 \left(\frac{4}{3} + 1 \right) = \frac{448}{81} \Longrightarrow 448 + 81 = \boxed{529}$$

Solution 4 (another version of Solution 3)

Taking the logarithm base $x$ of both sides, we arrive with:

$$y = \log_x y^x \Longrightarrow \frac{y}{x} = \log_{x} y = \log_x \left(\frac{3}{4}x\right) = \frac{3}{4}$$ Now we proceed by the logarithm rule $\log(ab)=\log a + \log b$. The equation becomes:

$$\log_x \frac{3}{4} + \log_x x = \frac{3}{4}$$ $$\Longleftrightarrow \log_x \frac{3}{4} + 1 = \frac{3}{4}$$ $$\Longleftrightarrow \log_x \frac{3}{4} = -\frac{1}{4}$$ $$\Longleftrightarrow x^{-\frac{1}{4}} = \frac{3}{4}$$ $$\Longleftrightarrow \frac{1}{x^{\frac{1}{4}}} = \frac{3}{4}$$ $$\Longleftrightarrow x^{\frac{1}{4}} = \frac{4}{3}$$ $$\Longleftrightarrow \sqrt[4]{x} = \frac{4}{3}$$ $$\Longleftrightarrow x = \left(\frac{4}{3}\right)^4=\frac{256}{81}$$ Then find $y$ as in solution 3, and we get $\boxed{529}$.