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Difference between revisions of "2021 AMC 10A Problems/Problem 1"
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<math>\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 12</math>
 
<math>\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 12</math>
  
−
== Solution ==
+
== Solution 1==
−
<math>(4-2)-(9-3)+(16-4)=2-6+12=8.</math> This corresponds to <math>\boxed{\text{D}}</math>
+
<math>(4-2)-(9-3)+(16-4)=2-6+12=8.</math> This corresponds to <math>\boxed{\textbf{(D) } 8}.</math>
  
 
-happykeeper
 
-happykeeper
−
==See also==
+
 
−
{{AMC10 box|year=2021|ab=A|num-b=2|num-a=4}}
+
== Solution 2==
 +
We have
 +
<cmath>\begin{align*}
 +
\left(2^2-2\right)-\left(3^2-3\right)+\left(4^2-4\right) &= 2(2-1)-3(3-1)+4(4-1) \\
 +
&= 2(1)-3(2)+4(3) \\
 +
&= 2-6+12 \\
 +
&= \boxed{\textbf{(D) } 8}.
 +
\end{align*}</cmath>
 +
~MRENTHUSIASM
 +
 
 +
==Solution 3 (Overkill: Just for Fun)==
 +
We have
 +
<cmath>\begin{align*}
 +
(2^2-2)-(3^2-3)+(4^2-4) &= 2^2+4^2-3^2-2+3-4 \\
 +
&=2^2+(4-3)(4+3)-3 \\
 +
&=2^2+7-3=2^2+4 \\
 +
&=4\cdot 2 \\
 +
&=\boxed{\textbf{(D) } 8}.
 +
\end{align*}</cmath>
 +
-[[User:PureSwag|PureSwag]]
 +
 
 +
==Solution 4 (When you have too much time) ==
 +
 
 +
Using the difference of squares, we have <cmath>(2-\sqrt{2})(2+\sqrt{2}) - (3-\sqrt{3})(3+\sqrt{3}) + (4-2)(4+2).</cmath>
 +
Knowing that <math>\sqrt{2} \approx 1.41</math> and <math>\sqrt{3} \approx 1.73,</math> we get <cmath>(2-\sqrt{2})(2+\sqrt{2}) - (3-\sqrt{3})(3+\sqrt{3}) + (4-2)(4+2) \approx 0.59\cdot 3.41 -1.26\cdot 4.73 + 2 \cdot 6
 +
=8.0521,</cmath> which is closest to <math>\boxed{\textbf{(D) } 8}.</math>
 +
 
 +
==Video Solution 1 (Lightning Fast)==
 +
https://youtu.be/cAMg15KhK6E
 +
 
 +
~ Education, the Study of everything
 +
 
 +
==Video Solution 2==
 +
https://youtu.be/cAMg15KhK6E
 +
 
 +
==Video Solution 3 (Very Very Quick Computation) ==
 +
https://www.youtube.com/watch?v=m0_UMI2mnZs&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=2
 +
~North America Mathematic Contest Go Go Go
 +
 
 +
==Video Solution 4 (Quick Computation) ==
 +
https://youtu.be/C3n2hgBhyXc?t=37
 +
~ThePuzzlr
 +
 
 +
== Video Solution 5 by OmegaLearn (Arithmetic Computation) ==
 +
https://youtu.be/0VvM_f-IDvE
 +
 
 +
~ pi_is_3.14
 +
 
 +
==Video Solution 6==
 +
https://youtu.be/4dkzuRHJieQ
 +
 
 +
~savannahsolver
 +
 
 +
==Video Solution 7==
 +
https://youtu.be/50CThrk3RcM
 +
 
 +
~IceMatrix
 +
 
 +
==Video Solution 8 (Problems 1-3)==
 +
https://youtu.be/CupJpUzKPB0
 +
 
 +
~MathWithPi
 +
 
 +
==Video Solution 9==
 +
https://youtu.be/slVBYmcDMOI
 +
 
 +
~TheLearningRoyal
 +
 
 +
== See Also ==
 +
{{AMC10 box|year=2021|ab=A|before=First Problem|num-a=2}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 19:22, 14 October 2024

Problem

What is the value of \[(2^2-2)-(3^2-3)+(4^2-4)\]

$\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 12$

Solution 1

$(4-2)-(9-3)+(16-4)=2-6+12=8.$ This corresponds to $\boxed{\textbf{(D) } 8}.$

-happykeeper

Solution 2

We have \begin{align*} \left(2^2-2\right)-\left(3^2-3\right)+\left(4^2-4\right) &= 2(2-1)-3(3-1)+4(4-1) \\ &= 2(1)-3(2)+4(3) \\ &= 2-6+12 \\ &= \boxed{\textbf{(D) } 8}. \end{align*} ~MRENTHUSIASM

Solution 3 (Overkill: Just for Fun)

We have \begin{align*} (2^2-2)-(3^2-3)+(4^2-4) &= 2^2+4^2-3^2-2+3-4 \\ &=2^2+(4-3)(4+3)-3 \\ &=2^2+7-3=2^2+4 \\ &=4\cdot 2 \\ &=\boxed{\textbf{(D) } 8}. \end{align*} -PureSwag

Solution 4 (When you have too much time)

Using the difference of squares, we have \[(2-\sqrt{2})(2+\sqrt{2}) - (3-\sqrt{3})(3+\sqrt{3}) + (4-2)(4+2).\] Knowing that $\sqrt{2} \approx 1.41$ and $\sqrt{3} \approx 1.73,$ we get \[(2-\sqrt{2})(2+\sqrt{2}) - (3-\sqrt{3})(3+\sqrt{3}) + (4-2)(4+2) \approx 0.59\cdot 3.41 -1.26\cdot 4.73 + 2 \cdot 6 =8.0521,\] which is closest to $\boxed{\textbf{(D) } 8}.$

Video Solution 1 (Lightning Fast)

https://youtu.be/cAMg15KhK6E

~ Education, the Study of everything

Video Solution 2

https://youtu.be/cAMg15KhK6E

Video Solution 3 (Very Very Quick Computation)

https://www.youtube.com/watch?v=m0_UMI2mnZs&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=2 ~North America Mathematic Contest Go Go Go

Video Solution 4 (Quick Computation)

https://youtu.be/C3n2hgBhyXc?t=37 ~ThePuzzlr

Video Solution 5 by OmegaLearn (Arithmetic Computation)

https://youtu.be/0VvM_f-IDvE

~ pi_is_3.14

Video Solution 6

https://youtu.be/4dkzuRHJieQ

~savannahsolver

Video Solution 7

https://youtu.be/50CThrk3RcM

~IceMatrix

Video Solution 8 (Problems 1-3)

https://youtu.be/CupJpUzKPB0

~MathWithPi

Video Solution 9

https://youtu.be/slVBYmcDMOI

~TheLearningRoyal

See Also

2021 AMC 10A (Problems • Answer Key • Resources)
Preceded by
First Problem 		Followed by
Problem 2
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 AMC 10 Problems and Solutions

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