2022 AMC 10B 第 1 题

Question

对所有实数

x

和

y

，定义

x~\diamondsuit~ y

为

|x-y|

。求

(1~\diamondsuit~(2~\diamondsuit~3))-((1~\diamondsuit~2)~\diamondsuit~3)\text{？}

Define

x~\diamondsuit~ y

to be

|x-y|

for all real numbers

x

and

y.

What is the value of

(1~\diamondsuit~(2~\diamondsuit~3))-((1~\diamondsuit~2)~\diamondsuit~3)?

-2

-1

0

1

2

这个运算表示两个数的差的绝对值。

Remember that the operation is absolute difference

从最里面的菱形运算开始计算。

Evaluate each diamond operation from the inside out

从最内层的运算开始，依次向外计算可得

\begin{aligned} 1\mathbin\diamondsuit(2\mathbin\diamondsuit3)&=|1-|2-3||=0,\\ (1\mathbin\diamondsuit2)\mathbin\diamondsuit3&=||1-2|-3|=2 \end{aligned}\text{。}

因此，题中所求的差为

0-2=-2

。

所以正确答案是 A。

Working from the innermost operations outward gives

\begin{aligned} 1\mathbin\diamondsuit(2\mathbin\diamondsuit3)&=|1-|2-3||=0,\\ (1\mathbin\diamondsuit2)\mathbin\diamondsuit3&=||1-2|-3|=2. \end{aligned}

Therefore, the given difference is

0-2=-2.

Thus, the answer is A .

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