Difference between revisions of "2022 AMC 8 Problems/Problem 2"
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a \, \bigstar \, b &= (a - b)^2 | a \, \bigstar \, b &= (a - b)^2 | ||
\end{align*}</cmath> | \end{align*}</cmath> | ||
| − | What is the | + | What is the output of <math>(5 \, \blacklozenge \, 3) \, \bigstar \, 6?</math> |
<math>\textbf{(A) } {-}20 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 100 \qquad \textbf{(E) } 220</math> | <math>\textbf{(A) } {-}20 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 100 \qquad \textbf{(E) } 220</math> | ||
==Solution== | ==Solution== | ||
| − | We | + | We can substitute <math>5</math>, <math>3</math>, and <math>6</math> into the functions' definitions: |
<cmath>\begin{align*} | <cmath>\begin{align*} | ||
(5 \, \blacklozenge \, 3) \, \bigstar \, 6 &= \left(5^2-3^2\right) \, \bigstar \, 6 \\ | (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 \, \bigstar \, 6 \\ | ||
&= (16-6)^2 \\ | &= (16-6)^2 \\ | ||
&= \boxed{\textbf{(D) } 100}. | &= \boxed{\textbf{(D) } 100}. | ||
\end{align*}</cmath> | \end{align*}</cmath> | ||
| − | <i>pog</i> | + | <i>~pog</i> ~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 | ~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== | ==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}} | ||
Latest revision as of 14:23, 23 November 2023
Contents
Problem
Consider these two operations:
What is the output of 
Solution
We can substitute 
, 
, and 
into the functions' definitions:
~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!!!)
~Education, the Study of Everything
Video Solution 3
https://www.youtube.com/watch?v=Ij9pAy6tQSg&t=91 ~Interstigation
Video Solution 4
~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) | ||
| Preceded by 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
