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Difference between revisions of "2021 AMC 12B Problems"

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==Problem 5==
 
==Problem 5==
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These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.
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The point <math>P(a,b)</math> in the <math>xy</math>-plane is first rotated counterclockwise by <math>90^\circ</math> around the point <math>(1,5)</math> and then reflected about the line <math>y=-x</math>. The image of <math>P</math> after these two transformations is at <math>(-6,3)</math>. What is <math>b-a?</math>
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<math>\textbf{(A) }1 \qquad \textbf{(B) }3 \qquad \textbf{(C) }5 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9</math>
  
 
[[2021 AMC 12B Problems/Problem 5|Solution]]
 
[[2021 AMC 12B Problems/Problem 5|Solution]]

Revision as of 15:46, 11 February 2021

2021 AMC 12B (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the test if before 2006. No problems on the test will require the use of a calculator).
  4. Figures are not necessarily drawn to scale.
  5. You will have 75 minutes working time to complete the test.
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

Problem 1

How many integer values of $x$ satisfy $|x|<3\pi?$

$\textbf{(A) }9 \qquad \textbf{(B) }10 \qquad \textbf{(C) }18 \qquad \textbf{(D) }19 \qquad \textbf{(E) }20$

Solution

Problem 2

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

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

Solution

Problem 3

Suppose\[2+\frac{1}{1+\frac{1}{2+\frac{2}{3+x}}}=\frac{144}{53}.\]What is the value of $x?$

$\textbf{(A) }\frac34 \qquad \textbf{(B) }\frac78 \qquad \textbf{(C) }\frac{14}{15} \qquad \textbf{(D) }\frac{37}{38} \qquad \textbf{(E) }\frac{52}{53}$

Solution

Problem 4

Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is $84$, and the afternoon class's mean score is $70$. The ratio of the number of students in the morning clas to the number of students in the afternoon class is $\frac34$. What is the mean of the score of all the students?

$\textbf{(A) }74 \qquad \textbf{(B) }75 \qquad \textbf{(C) }76 \qquad \textbf{(D) }77 \qquad \textbf{(E) }78$

Solution

Problem 5

The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^\circ$ around the point $(1,5)$ and then reflected about the line $y=-x$. The image of $P$ after these two transformations is at $(-6,3)$. What is $b-a?$

$\textbf{(A) }1 \qquad \textbf{(B) }3 \qquad \textbf{(C) }5 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9$

Solution

Problem 6

These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.

Solution

Problem 7

These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.

Solution

Problem 8

These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.

Solution

Problem 9

These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.

Solution

Problem 10

These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.

Solution

Problem 11

These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.

Solution

Problem 12

These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.

Solution

Problem 13

These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.

Solution

Problem 14

These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.

Solution

Problem 15

These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.

Solution

Problem 16

These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.

Solution

Problem 17

These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.

Solution

Problem 18

These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.

Solution

Problem 19

These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.

Solution

Problem 20

These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.

Solution

Problem 21

These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.

Solution

Problem 22

These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.

Solution

Problem 23

These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.

Solution

Problem 24

These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.

Solution

Problem 25

These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.

Solution

See also

2021 AMC 12B (Problems • Answer Key • Resources)
Preceded by
2021 AMC 12A Problems 		Followed by
2022 AMC 12A Problems
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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