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

Latest revision as of 00:04, 17 October 2024

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 1

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)

Solution 2

We can find a general solution to any $((a \, \blacklozenge \, b) \, \bigstar \, c)$ . $((a \, \blacklozenge \, b) \, \bigstar \, c)$ $=((a^2-b^2) \, \bigstar \, c)$ $=(a^2-b^2-c)^2$ $=a^4+b^4-(a^2)(b^2)-2(a^2)(c)-(b^2)(a^2)+2(b^2)(c)+c^2$ $=5^4+3^4-(5^2)(3^2)-2(5^2)(6)-(3^2)(5^2)+2(3^2)(6)+6^2$ $=625+81-225-300-225+108+36$ $=\boxed{\textbf{(D) } 100}$

~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 ~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

@@ Line 10: / Line 10: @@
 <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 can substitute <math>5</math>, <math>3</math>, and <math>6</math> into the functions' definitions:
 <cmath>\begin{align*}
@@ Line 21: / Line 21: @@
 <i>~pog</i> ~MathFun1000 (Minor Edits)
-==Video Solution 1==
+==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
 ~Interstigation
-==Video Solution 2==
+==Video Solution 4==
 https://youtu.be/YYuEBGoEK1Y
 ~savannahsolver
-==Video Solution 3==
+==Video Solution 5==
 https://youtu.be/Q0R6dnIO95Y?t=53
 ~STEMbreezy
+==Video Solution 6==
+https://www.youtube.com/watch?v=c123kPqd11I
+~harungurcan
 ==See Also==
 {{AMC8 box|year=2022|num-b=1|num-a=3}}
 {{MAA Notice}}

2022 AMC 8 (Problems • Answer Key • Resources)
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