Difference between revisions of "2021 AMC 12B Problems/Problem 2"

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The 2021 AMC 12B will be held on February 10th, 2021. The problems will not be made public until 24 hours after that.
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{{duplicate|[[2021 AMC 10B Problems#Problem 4|2021 AMC 10B #4]] and [[2021 AMC 12B Problems#Problem 2|2021 AMC 12B #2]]}}
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==Problem==
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At a math contest, <math>57</math> students are wearing blue shirts, and another <math>75</math> students are wearing yellow shirts. The <math>132</math> students are assigned into <math>66</math> pairs. In exactly <math>23</math> of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts?
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<math>\textbf{(A)} ~23 \qquad\textbf{(B)} ~32 \qquad\textbf{(C)} ~37 \qquad\textbf{(D)} ~41 \qquad\textbf{(E)} ~64</math>
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==Solution==
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There are <math>46</math> students paired with a blue partner. The other <math>11</math> students wearing blue shirts must each be paired with a partner wearing a shirt of the opposite color. There are <math>64</math> students remaining. Therefore the requested number of pairs is <math>\tfrac{64}{2}=\boxed{\textbf{(B)} ~32}</math> ~Punxsutawney Phil
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==Video Solution by Punxsutawney Phil==
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https://youtube.com/watch?v=qpvS2PVkI8A&t=55s
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== Video Solution by OmegaLearn (System of Equations) ==
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https://youtu.be/hyYg62tT0sY
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~ pi_is_3.14
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==Video Solution by Hawk Math==
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https://www.youtube.com/watch?v=VzwxbsuSQ80
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==Video Solution by TheBeautyofMath==
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https://youtu.be/gLahuINjRzU?t=626
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https://youtu.be/EMzdnr1nZcE?t=154
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~IceMatrix
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==Video Solution by Interstigation==
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https://youtu.be/DvpN56Ob6Zw?t=286
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~Interstigation
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==Video Solution by WhyMath==
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https://youtu.be/JOc-5A9Q9Ic
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~savannahsolver
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==Video Solution (Under 2 min!)==
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https://youtu.be/5yCyXUnTu5Y
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~Education, the Study of Everything
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==See Also==
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{{AMC12 box|year=2021|ab=B|num-b=1|num-a=3}}
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{{AMC10 box|year=2021|ab=B|num-b=3|num-a=5}}
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{{MAA Notice}}

Latest revision as of 22:49, 18 July 2023

The following problem is from both the 2021 AMC 10B #4 and 2021 AMC 12B #2, so both problems redirect to this page.

Problem

At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts?

$\textbf{(A)} ~23 \qquad\textbf{(B)} ~32 \qquad\textbf{(C)} ~37 \qquad\textbf{(D)} ~41 \qquad\textbf{(E)} ~64$

Solution

There are $46$ students paired with a blue partner. The other $11$ students wearing blue shirts must each be paired with a partner wearing a shirt of the opposite color. There are $64$ students remaining. Therefore the requested number of pairs is $\tfrac{64}{2}=\boxed{\textbf{(B)} ~32}$ ~Punxsutawney Phil

Video Solution by Punxsutawney Phil

https://youtube.com/watch?v=qpvS2PVkI8A&t=55s

Video Solution by OmegaLearn (System of Equations)

https://youtu.be/hyYg62tT0sY

~ pi_is_3.14

Video Solution by Hawk Math

https://www.youtube.com/watch?v=VzwxbsuSQ80

Video Solution by TheBeautyofMath

https://youtu.be/gLahuINjRzU?t=626

https://youtu.be/EMzdnr1nZcE?t=154

~IceMatrix

Video Solution by Interstigation

https://youtu.be/DvpN56Ob6Zw?t=286

~Interstigation

Video Solution by WhyMath

https://youtu.be/JOc-5A9Q9Ic

~savannahsolver

Video Solution (Under 2 min!)

https://youtu.be/5yCyXUnTu5Y

~Education, the Study of Everything

See Also

2021 AMC 12B (ProblemsAnswer KeyResources)
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 AMC 12 Problems and Solutions
2021 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 3
Followed by
Problem 5
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. AMC logo.png