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 1== |
| − | 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== | + | ==Solution 2== |
| + | We can find a general solution to any <math>((a \, \blacklozenge \, b) \, \bigstar \, c)</math>. | ||
| + | <cmath> ((a \, \blacklozenge \, b) \, \bigstar \, c)</cmath> | ||
| + | <cmath> =((a^2-b^2) \, \bigstar \, c) </cmath> | ||
| + | <cmath> =(a^2-b^2-c)^2 </cmath> | ||
| + | <cmath> =a^4+b^4-(a^2)(b^2)-2(a^2)(c)-(b^2)(a^2)+2(b^2)(c)+c^2 </cmath> | ||
| + | <cmath> =5^4+3^4-(5^2)(3^2)-2(5^2)(6)-(3^2)(5^2)+2(3^2)(6)+6^2 </cmath> | ||
| + | <cmath> =625+81-225-300-225+108+36 </cmath> | ||
| + | <cmath> =\boxed{\textbf{(D) } 100} </cmath> | ||
| + | |||
| + | ~megaboy6679 | ||
| + | |||
| + | ==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 | https://www.youtube.com/watch?v=Ij9pAy6tQSg&t=91 | ||
~Interstigation | ~Interstigation | ||
| − | ==Video Solution | + | ==Video Solution 4== |
https://youtu.be/YYuEBGoEK1Y | https://youtu.be/YYuEBGoEK1Y | ||
~savannahsolver | ~savannahsolver | ||
| − | ==Video Solution== | + | ==Video Solution 5== |
https://youtu.be/Q0R6dnIO95Y?t=53 | https://youtu.be/Q0R6dnIO95Y?t=53 | ||
~STEMbreezy | ~STEMbreezy | ||
| + | |||
| + | ==Video Solution 6== | ||
| + | https://www.youtube.com/watch?v=c123kPqd11I | ||
| + | |||
| + | ~harungurcan | ||
| + | |||
| + | ==Video Solution 7 by Dr. David== | ||
| + | |||
| + | https://youtu.be/ItPIuHdgyNk | ||
==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 01:47, 19 November 2024
Contents
Problem
Consider these two operations:
What is the output of 
Solution 1
We can substitute 
, 
, and 
into the functions' definitions:
~pog ~MathFun1000 (Minor Edits)
Solution 2
We can find a general solution to any 
.
![\[((a \, \blacklozenge \, b) \, \bigstar \, c)\]](http://latex.artofproblemsolving.com/9/b/1/9b1ead4edbaae012d71fc60a0fb5f97d60ea4a95.png)
![\[=((a^2-b^2) \, \bigstar \, c)\]](http://latex.artofproblemsolving.com/1/3/0/130283c842ecd4a186bfc2a3a91da5d10b49d92e.png)
![\[=(a^2-b^2-c)^2\]](http://latex.artofproblemsolving.com/0/c/a/0ca0fb3ae428092c4149b4c4afdd2bb3bb2306e0.png)
![\[=a^4+b^4-(a^2)(b^2)-2(a^2)(c)-(b^2)(a^2)+2(b^2)(c)+c^2\]](http://latex.artofproblemsolving.com/0/0/d/00dc98219662e298a83f59b9e87e169d34ef4dc5.png)
![\[=5^4+3^4-(5^2)(3^2)-2(5^2)(6)-(3^2)(5^2)+2(3^2)(6)+6^2\]](http://latex.artofproblemsolving.com/4/e/d/4edeffd78b5b1feb3e9fdf7af355cff8ed2829a1.png)
![\[=625+81-225-300-225+108+36\]](http://latex.artofproblemsolving.com/2/3/5/2358711b9bc73279606f6b4824474e91e457133a.png)
![\[=\boxed{\textbf{(D) } 100}\]](http://latex.artofproblemsolving.com/c/a/d/cad27560c19ca35edc6eeac0171d9866b590ad60.png)
~megaboy6679
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
Video Solution 7 by Dr. David
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. 
