2022 AMC 10B Problem 1

Question

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

Remember that the operation is absolute difference

Evaluate each diamond operation from the inside out

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 .

Problem 1 in other years