2022 AMC 12B Problem 1

Problem 1 of 25EasierAlgebra

Define

x \diamond y

to be

|x - y|

for all real numbers

x

and

y.

What is the value of

(1 \diamond (2 \diamond 3)) - ((1 \diamond 2) \diamond 3)?

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Solution

Since

2 \diamond 3 = |2-3| = 1,

we get

1 \diamond (2 \diamond 3) = 1 \diamond 1 = 0.

Since

1 \diamond 2 = |1-2| = 1,

we get

(1 \diamond 2) \diamond 3 = 1 \diamond 3 = |1-3| = 2.

The value is

0 - 2 = -2.

Thus, the correct answer is A.

Concepts: custom operation · absolute value · order of operations

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.