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

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 $a\diamond b=\dfrac{a}{b}$ 。

因此所以 $x=\dfrac{100}{336}=\dfrac{25}{84}$ ，且 $p+q=25+84=109$ 。 $\begin{gathered} 2016\diamond(6\diamond x)\\ =2016\diamond\dfrac{6}{x}\\ =\dfrac{2016}{6/x}\\ =336x=100, \end{gathered}$

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

解答：

其他年份的第 20 题