Question

对任意三个实数 $a$ 、 $b$ 、 $c$ ，其中 $b \neq c$ ，定义运算 $\diamond$ 为求 $\diamond(\diamond(1, 2, 3), \diamond(2, 3, 1), \diamond(3, 1, 2))$ 的值。 $\diamond(a, b, c) = \dfrac{a}{b - c}.$

For any three real numbers $a,$ $b,$ and $c,$ with $b \neq c,$ the operation $\diamond$ is defined by $\diamond(a, b, c) = \dfrac{a}{b - c}.$ What is $\diamond(\diamond(1, 2, 3), \diamond(2, 3, 1), \diamond(3, 1, 2))?$

$-\dfrac{1}{2}$

$-\dfrac{1}{4}$

$0$

$\dfrac{1}{4}$

$\dfrac{1}{2}$

答案：B

知识点：自定义运算分数

难度评级：980

解答：

三个内层表达式的值为 $\begin{aligned} \diamond(1,2,3) &= \dfrac{1}{2-3} = -1, \\ \diamond(2,3,1) &= \dfrac{2}{3-1} = 1, \\ \diamond(3,1,2) &= \dfrac{3}{1-2} = -3. \end{aligned}$

因此 $\begin{aligned} \diamond(-1, 1, -3) &= \dfrac{-1}{1 - (-3)} \\ &= -\dfrac{1}{4}. \end{aligned}$

所以正确答案是 B。

The inner values are $\begin{aligned} \diamond(1,2,3) &= \dfrac{1}{2-3} = -1, \\ \diamond(2,3,1) &= \dfrac{2}{3-1} = 1, \\ \diamond(3,1,2) &= \dfrac{3}{1-2} = -3. \end{aligned}$

Therefore $\begin{aligned} \diamond(-1, 1, -3) &= \dfrac{-1}{1 - (-3)} \\ &= -\dfrac{1}{4}. \end{aligned}$

Thus, the correct answer is B.

2004 AMC 10A 第 2 题

答案：B

难度评级：980

解答：

其他年份的第 2 题