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{\ | + | <math>(4-2)-(9-3)+(16-4)=2-6+12=8.</math> This corresponds to <math>\boxed{\textbf{(D) } 8}.</math> |
-happykeeper | -happykeeper | ||
| + | |||
| + | == 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
Contents
- 1 Problem
- 2 Solution 1
- 3 Solution 2
- 4 Solution 3 (Overkill: Just for Fun)
- 5 Solution 4 (When you have too much time)
- 6 Video Solution 1 (Lightning Fast)
- 7 Video Solution 2
- 8 Video Solution 3 (Very Very Quick Computation)
- 9 Video Solution 4 (Quick Computation)
- 10 Video Solution 5 by OmegaLearn (Arithmetic Computation)
- 11 Video Solution 6
- 12 Video Solution 7
- 13 Video Solution 8 (Problems 1-3)
- 14 Video Solution 9
- 15 See Also
Problem
What is the value of ![\[(2^2-2)-(3^2-3)+(4^2-4)\]](http://latex.artofproblemsolving.com/f/b/f/fbf9a3889b146c23eb653b9dc16255c03a229dfe.png)
Solution 1
This corresponds to 
-happykeeper
Solution 2
We have
~MRENTHUSIASM
Solution 3 (Overkill: Just for Fun)
We have
-PureSwag
Solution 4 (When you have too much time)
Using the difference of squares, we have ![\[(2-\sqrt{2})(2+\sqrt{2}) - (3-\sqrt{3})(3+\sqrt{3}) + (4-2)(4+2).\]](http://latex.artofproblemsolving.com/d/c/e/dce654c8a197d06312c02b9cf3b07066856347c3.png)
Knowing that 
and 
we get ![\[(2-\sqrt{2})(2+\sqrt{2}) - (3-\sqrt{3})(3+\sqrt{3}) + (4-2)(4+2) \approx 0.59\cdot 3.41 -1.26\cdot 4.73 + 2 \cdot 6 =8.0521,\]](http://latex.artofproblemsolving.com/9/2/c/92c78ba7e8054466f98fc098f06c66d0c2ab6367.png)
which is closest to
Video Solution 1 (Lightning Fast)
~ Education, the Study of everything
Video Solution 2
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)
~ pi_is_3.14
Video Solution 6
~savannahsolver
Video Solution 7
~IceMatrix
Video Solution 8 (Problems 1-3)
~MathWithPi
Video Solution 9
~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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
