2010 AIME I 第 3 题

先试着解答 2010 AIME I 第 3 题，然后核对你的答案与精心整理的解答，解答来自 LIVE by Po-Shen Loh。你也可以参加完整限时模拟考试、查看全部 2010 AIME I 解答，或核对答案。

所有题目均经美国数学协会（MAA）官方合法授权使用。

3.

假设

y = \frac{3}{4}x

且

x^y = y^x

。量

x + y

可表示为有理数

\frac{r}{s}

，其中

r

和

s

是互质的正整数。求

r + s

。

Suppose that

y = \frac{3}{4}x

and

x^y = y^x.

The quantity

x + y

can be expressed as a rational number

\frac{r}{s},

where

r

and

s

are relatively prime positive integers. Find

r + s.

答案

答案：529

知识点：指数换元法

难度评级：2230

小提示：

将

y = \frac{3}{4}x

代入

x^y = y^x

，再对两边取

x

次方根

Substitute

y = \frac{3}{4}x

into

x^y = y^x

and take

x

th roots of both sides

大提示：

方程变为

x^{\frac{3}{4}} = \frac{3}{4}x

，所以

x^{-\frac{1}{4}} = \frac{3}{4}

，从而

x = \left(\frac{4}{3}\right)^4

The equation becomes

x^{\frac{3}{4}} = \frac{3}{4}x,

so

x^{-\frac{1}{4}} = \frac{3}{4},

which gives

x = \left(\frac{4}{3}\right)^4

解答：

将

y = \frac{3}{4}x

代入

x^y = y^x

，得

x^{\frac{3}{4}x} = \left(\tfrac{3}{4}x\right)^{x}\text{。}

对两边取

x

次方根（这里各量为正），得

x^{\frac{3}{4}} = \frac{3}{4}x

，所以除以

x

得

x^{-\frac{1}{4}} = \frac{3}{4}

，也就是

x = \left(\frac{4}{3}\right)^4 = \frac{256}{81}

。

于是

y = \frac{3}{4} \cdot \frac{256}{81} = \frac{64}{27}

，并且

x + y = \frac{256}{81} + \frac{192}{81} = \frac{448}{81}\text{。}

因为

\gcd(448, 81) = 1

，答案为

448 + 81 = 529

。

Substituting

y = \frac{3}{4}x

into

x^y = y^x

gives

x^{\frac{3}{4}x} = \left(\tfrac{3}{4}x\right)^{x}.

Taking

x

th roots (the quantities here are positive),

x^{\frac{3}{4}} = \frac{3}{4}x,

so dividing by

x

yields

x^{-\frac{1}{4}} = \frac{3}{4},

that is,

x = \left(\frac{4}{3}\right)^4 = \frac{256}{81}.

Then

y = \frac{3}{4} \cdot \frac{256}{81} = \frac{64}{27},

and

x + y = \frac{256}{81} + \frac{192}{81} = \frac{448}{81}.

Since

\gcd(448, 81) = 1,

the answer is

448 + 81 = 529.