Difference between revisions of "2022 AMC 10B Problems/Problem 1"
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==Problem== | ==Problem== | ||
− | Define <math>x\ | + | Define <math>x\diamond y</math> to be <math>|x-y|</math> for all real numbers <math>x</math> and <math>y.</math> What is the value of <cmath>(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?</cmath> |
− | |||
− | == Solution == | + | <math> \textbf{(A)}\ {-}2 \qquad |
+ | \textbf{(B)}\ {-}1 \qquad | ||
+ | \textbf{(C)}\ 0 \qquad | ||
+ | \textbf{(D)}\ 1 \qquad | ||
+ | \textbf{(E)}\ 2</math> | ||
+ | |||
+ | == Solution 1 == | ||
We have <cmath>\begin{align*} | We have <cmath>\begin{align*} | ||
− | (1\ | + | (1\diamond(2\diamond3))-((1\diamond2)\diamond3) &= |1-|2-3|| - ||1-2|-3| \\ |
&= |1-1| - |1-3| \\ | &= |1-1| - |1-3| \\ | ||
&= 0-2 \\ | &= 0-2 \\ | ||
− | &=\boxed{\textbf{(A)}\ {-}2}. | + | &= \boxed{\textbf{(A)}\ {-}2}. |
\end{align*}</cmath> | \end{align*}</cmath> | ||
~MRENTHUSIASM | ~MRENTHUSIASM | ||
+ | |||
+ | == 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{\textbf{(A)}\ {-}2}.</math> | ||
+ | |||
+ | ~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 | ||
== See Also == | == See Also == |
Latest revision as of 04:22, 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.
Contents
Problem
Define to be
for all real numbers
and
What is the value of
Solution 1
We have
~MRENTHUSIASM
Solution 2
Observe that the function is simply the positive difference between two numbers. Thus, we evaluate: the difference between
and
is
the difference between
and
is
the difference between
and
is
the difference between
and
is
and finally,
~Technodoggo
Video Solution (⚡️Solved in 50 seconds⚡️)
~Education, the Study of Everything
Video Solution(1-16)
~~Hayabusa1
Video Solution by Interstigation
See Also
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.