Difference between revisions of "2022 AMC 8 Problems/Problem 2"
(Created page with "==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 value of <math>(5 \,...") |
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Consider these two operations: | Consider these two operations: | ||
− | \begin{align*} | + | <cmath>\begin{align*} |
a \, \blacklozenge \, b &= a^2 - b^2\\ | a \, \blacklozenge \, b &= a^2 - b^2\\ | ||
a \, \bigstar \, b &= (a - b)^2 | a \, \bigstar \, b &= (a - b)^2 | ||
− | \end{align*} | + | \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== | ||
+ | We can substitute <math>5</math>, <math>3</math>, and <math>6</math> into the functions' definitions: | ||
+ | <cmath>\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*}</cmath> | ||
+ | <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 | ||
+ | |||
+ | ==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.