2016 AMC 12A 第 20 题

Question

二元运算

\diamond

满足：对所有非零实数

a

、

b

、

c

，

a\diamond(b\diamond c)=(a\diamond b)\cdot c

，且

a\diamond a=1

。（这里的点

\cdot

表示通常的乘法。）方程

2016\diamond(6\diamond x)=100

的解可写为

\dfrac{p}{q}

，其中

p

、

q

是互质正整数。

p+q

是多少？

A binary operation

\diamond

has the properties that

a\diamond(b\diamond c)=(a\diamond b)\cdot c

and that

a\diamond a=1

for all nonzero real numbers

a,

b,

and

c.

(Here the dot

\cdot

represents the usual multiplication operation.) The solution to the equation

2016\diamond(6\diamond x)=100

can be written as

\dfrac{p}{q},

where

p

and

q

are relatively prime positive integers. What is

p+q?

109

201

301

3049

33{,}601

答案：A

知识点：自定义运算函数方程

难度评级：1910

小提示：

用

a\diamond a=1

化简

a\diamond(a\diamond a)

。

Use

a\diamond a=1

to simplify

a\diamond(a\diamond a)

大提示：

证明

a\diamond b=\dfrac{a}{b}

，于是

2016\diamond(6\diamond x)=\dfrac{2016}{\frac{6}{x}}

。

Show

a\diamond b=\dfrac{a}{b},

so

2016\diamond(6\diamond x)=\dfrac{2016}{\frac{6}{x}}

解答：

令

b=c=a

，得到

a\diamond 1=a\diamond(a\diamond a)

=(a\diamond a)\cdot a=a

。再令

c=b

，得到

a=a\diamond 1

=a\diamond(b\diamond b)=(a\diamond b)\cdot b

，所以

a\diamond b=\dfrac{a}{b}

。

因此

\begin{gathered} 2016\diamond(6\diamond x)\\ =2016\diamond\dfrac{6}{x}\\ =\dfrac{2016}{\frac{6}{x}}\\ =336x=100 \end{gathered}\text{，}

所以

x=\dfrac{100}{336}=\dfrac{25}{84}

，且

p+q=25+84=109

。

所以正确答案是 A。

Setting

b=c=a

gives

a\diamond 1=a\diamond(a\diamond a)

=(a\diamond a)\cdot a=a.

Then setting

c=b

gives

a=a\diamond 1

=a\diamond(b\diamond b)=(a\diamond b)\cdot b,

so

a\diamond b=\dfrac{a}{b}.

Therefore

\begin{gathered} 2016\diamond(6\diamond x)\\ =2016\diamond\dfrac{6}{x}\\ =\dfrac{2016}{\frac{6}{x}}\\ =336x=100, \end{gathered}

so

x=\dfrac{100}{336}=\dfrac{25}{84}

and

p+q=25+84=109.

Thus, the correct answer is A.

2016 AMC 12A 第 20 题

答案：A

难度评级：1910

小提示：

大提示：

解答：

其他年份的第 20 题