2022 AMC 8 Problems/Problem 2: Difference between revisions

Revision as of 13:50, 11 August 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 (HOW TO THINK CREATIVELY!!!)

https://youtu.be/ytDV0GNc9Mw

~Education, the Study of Everything

Video Solution 1

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

Video Solution 2

https://youtu.be/YYuEBGoEK1Y

~savannahsolver

Video Solution 3

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

~STEMbreezy

Video Solution 4

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

~harungurcan

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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: Unable to save thumbnail to destination

@@ Line 40: / Line 40: @@
 ~STEMbreezy
+==Video Solution 4==
+https://www.youtube.com/watch?v=c123kPqd11I
+~harungurcan
 ==See Also==
 {{AMC8 box|year=2022|num-b=1|num-a=3}}
 {{MAA Notice}}

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