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Difference between revisions of "2022 AMC 8 Problems/Problem 2"

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<i>~pog</i> ~MathFun1000 (Minor Edits)
 
<i>~pog</i> ~MathFun1000 (Minor Edits)
  
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==Video Solution==
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==Video Solution 1 by Math-X (First understand the problem!!!)==
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https://youtu.be/oUEa7AjMF2A?si=JlQdlwkdbIYFNJXj&t=123
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~Math-X
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==Video Solution 2 (HOW TO THINK CREATIVELY!!!)==
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https://youtu.be/ytDV0GNc9Mw
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~Education, the Study of Everything
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==Video Solution 3==
 
https://www.youtube.com/watch?v=Ij9pAy6tQSg&t=91
 
https://www.youtube.com/watch?v=Ij9pAy6tQSg&t=91
 
~Interstigation
 
~Interstigation
  
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==Video Solution 2==
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==Video Solution 4==
 
https://youtu.be/YYuEBGoEK1Y
 
https://youtu.be/YYuEBGoEK1Y
  
 
~savannahsolver
 
~savannahsolver
  
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==Video Solution==
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==Video Solution 5==
 
https://youtu.be/Q0R6dnIO95Y?t=53
 
https://youtu.be/Q0R6dnIO95Y?t=53
  
 
~STEMbreezy
 
~STEMbreezy
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==Video Solution 6==
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https://www.youtube.com/watch?v=c123kPqd11I
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~harungurcan
  
 
==See Also==  
 
==See Also==  
 
{{AMC8 box|year=2022|num-b=1|num-a=3}}
 
{{AMC8 box|year=2022|num-b=1|num-a=3}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 13:23, 23 November 2023

Problem

Consider these two operations: \begin{align*} a \, \blacklozenge \, b &= a^2 - b^2\\ a \, \bigstar \, b &= (a - b)^2 \end{align*} What is the output of $(5 \, \blacklozenge \, 3) \, \bigstar \, 6?$

$\textbf{(A) } {-}20 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 100 \qquad \textbf{(E) } 220$

Solution

We can substitute $5$, $3$, and $6$ into the functions' definitions: \begin{align*} (5 \, \blacklozenge \, 3) \, \bigstar \, 6 &= \left(5^2-3^2\right) \, \bigstar \, 6 \\ (5 \, \blacklozenge \, 3) \, \bigstar \, 6 &= \left(25-9\right) \, \bigstar \, 6 \\ &= 16 \, \bigstar \, 6 \\ &= (16-6)^2 \\ &= \boxed{\textbf{(D) } 100}. \end{align*} ~pog ~MathFun1000 (Minor Edits)

Video Solution 1 by Math-X (First understand the problem!!!)

https://youtu.be/oUEa7AjMF2A?si=JlQdlwkdbIYFNJXj&t=123

~Math-X

Video Solution 2 (HOW TO THINK CREATIVELY!!!)

https://youtu.be/ytDV0GNc9Mw

~Education, the Study of Everything

Video Solution 3

https://www.youtube.com/watch?v=Ij9pAy6tQSg&t=91 ~Interstigation

Video Solution 4

https://youtu.be/YYuEBGoEK1Y

~savannahsolver

Video Solution 5

https://youtu.be/Q0R6dnIO95Y?t=53

~STEMbreezy

Video Solution 6

https://www.youtube.com/watch?v=c123kPqd11I

~harungurcan

See Also

2022 AMC 8 (Problems • Answer Key • Resources)
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Problem 1 		Followed by
Problem 3
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 AJHSME/AMC 8 Problems and Solutions

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