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2022 AMC 10B Problem 1

Problem 1 of 25EasierAlgebra

Define x ♢ yx~\diamondsuit~ y to be ∣x−y∣|x-y| for all real numbers xx and y.y. What is the value of (1 ♢ (2 ♢ 3))−((1 ♢ 2) ♢ 3)?(1~\diamondsuit~(2~\diamondsuit~3))-((1~\diamondsuit~2)~\diamondsuit~3)?

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Solution

Working from the innermost operations outward gives 1♢(2♢3)=∣1−∣2−3∣∣=0,(1♢2)♢3=∣∣1−2∣−3∣=2. \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=−2.0-2=-2. Thus, the answer is A .

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Concepts: custom operation · absolute value

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