Difference between revisions of "2022 AMC 10B Problems/Problem 1"

Revision as of 01:36, 23 October 2023

The following problem is from both the 2022 AMC 10B #1 and 2022 AMC 12B #1, so both problems redirect to this page.

Problem

Define $x\diamond y$ to be $|x-y|$ for all real numbers $x$ and $y.$ What is the value of $(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?$

$\textbf{(A)}\ {-}2 \qquad \textbf{(B)}\ {-}1 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 2$

Solution

We have $\begin{align*} (1\diamond(2\diamond3))-((1\diamond2)\diamond3) &= |1-|2-3|| - ||1-2|-3| \\ &= |1-1| - |1-3| \\ &= 0-2 \\ &=\boxed{\textbf{(A)}\ {-}2}. \end{align*}$ ~MRENTHUSIASM

Solution 2

Observe that the $\diamond$ function is simply the positive difference between two numbers. Thus, we evaluate: the difference between $2$ and $3$ is $1$ ; the difference between $1$ and $1$ is $0$ ; the difference between $1$ and $2$ is $1$ ; the difference between $1$ and $3$ is $2$ ; and finally, $0-2=\boxed{\textbf{(A)}\ {-}2}.$

~Technodoggo

Video Solution (⚡️Solved in 50 seconds⚡️)

https://youtu.be/4xOeX0aQF3U

~Education, the Study of Everything

Video Solution(1-16)

https://youtu.be/SCwQ9jUfr0g

~~Hayabusa1

Video Solution by Interstigation

https://youtu.be/_KNR0JV5rdI

2022 AMC 10B (Problems • Answer Key • Resources)
Preceded by First Problem	Followed by Problem 2
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

2022 AMC 12B (Problems • Answer Key • Resources)
Preceded by First Problem	Followed by Problem 2
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 20: / Line 20: @@
 == Solution 2 ==
-Observe that the <math>\diamond</math> function is simply the positive difference between two numbers. Thus, we evaluate: the difference between <math>2</math> and <math>3</math> is <math>1</math>; the difference between <math>1</math> and <math>1</math> is <math>0</math>; the difference between <math>1</math> and <math>2</math> is <math>1</math>; the difference between <math>1</math> and <math>3</math> is <math>2</math>; and finally, <math>0-2=\boxed{\text{(A)}~-2}.</math>
+Observe that the <math>\diamond</math> function is simply the positive difference between two numbers. Thus, we evaluate: the difference between <math>2</math> and <math>3</math> is <math>1</math>; the difference between <math>1</math> and <math>1</math> is <math>0</math>; the difference between <math>1</math> and <math>2</math> is <math>1</math>; the difference between <math>1</math> and <math>3</math> is <math>2</math>; and finally, <math>0-2=\boxed{\textbf{(A)}\ {-}2}.</math>
 ~Technodoggo

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