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Difference between revisions of "2021 CMC 12A Problems"

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{{CMC12 Problems|ab=A|year=2021}}
 
{{CMC12 Problems|ab=A|year=2021}}
 
==Problem 1==
 
==Problem 1==
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What is the value of <cmath>\frac{21^2+\tfrac{21}{20}}{20^2+\tfrac{20}{21}}?</cmath>
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<math>\textbf{(A) }\tfrac{400}{441}\qquad\textbf{(B) }\tfrac{20}{21}\qquad\textbf{(C) } 1\qquad\textbf{(D) }\tfrac{21}{20}\qquad\textbf{(E) }\tfrac{441}{400}\qquad</math>
  
 
[[2021 CMC 12A Problems/Problem 1|Solution]]
 
[[2021 CMC 12A Problems/Problem 1|Solution]]
 
==Problem 2==
 
==Problem 2==
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Hold your horses! The test hasn't finished yet.
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Two circles of equal radius <math>r</math> have an overlap area of <math>7\pi</math> and the total area covered by the circles is <math>25\pi</math>. What is the value of <math>r</math>?
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<math>\textbf{(A) } 2\sqrt{2}\qquad\textbf{(B) } 2\sqrt{3}\qquad\textbf{(C) } 4\qquad\textbf{(D) } 5\qquad\textbf{(E) } 4\sqrt{2}\qquad</math>
  
 
[[2021 CMC 12A Problems/Problem 2|Solution]]
 
[[2021 CMC 12A Problems/Problem 2|Solution]]
 
==Problem 3==
 
==Problem 3==
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Hold your horses! The test hasn't finished yet.
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A pyramid whose base is a regular <math>n</math>-gon has the same number of edges as a prism whose base is a regular <math>m</math>-gon. What is the smallest possible value of <math>n</math>?
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<math>\textbf{(A) } 3\qquad\textbf{(B) } 4\qquad\textbf{(C) } 6\qquad\textbf{(D) } 9\qquad\textbf{(E) } 12\qquad</math>
  
 
[[2021 CMC 12A Problems/Problem 3|Solution]]
 
[[2021 CMC 12A Problems/Problem 3|Solution]]
 
==Problem 4==
 
==Problem 4==
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Hold your horses! The test hasn't finished yet.
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There exists a positive integer <math>N</math> such that <cmath>\frac{\tfrac{1}{999}+\tfrac{1}{1001}}{2}=\frac{1}{1000}+\frac{1}{N}</cmath> What is the sum of the digits of <math>N</math>?
 +
 
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<math>\textbf{(A) } 36\qquad\textbf{(B) } 45\qquad\textbf{(C) } 54\qquad\textbf{(D) } 63\qquad\textbf{(E) } 72\qquad</math>
  
 
[[2021 CMC 12A Problems/Problem 4|Solution]]
 
[[2021 CMC 12A Problems/Problem 4|Solution]]
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==Problem 5==
 
==Problem 5==
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Hold your horses! The test hasn't finished yet.
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For positive integers <math>m>1</math> and <math>n>1,</math> the sum of the first <math>m</math> multiples of <math>n</math> is <math>2020</math>. Compute <math>m+n</math>.
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<math>\textbf{(A) } 206\qquad\textbf{(B) } 208\qquad\textbf{(C) } 210\qquad\textbf{(D) } 212\qquad\textbf{(E) } 214\qquad</math>
  
 
[[2021 CMC 12A Problems/Problem 5|Solution]]
 
[[2021 CMC 12A Problems/Problem 5|Solution]]
 
==Problem 6==
 
==Problem 6==
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Hold your horses! The test hasn't finished yet.
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How many of the following statements are true for every parallelogram <math>\mathcal{P}</math>?
 +
        i. The perpendicular bisectors of the sides of <math>\mathcal{P}</math> all share at least one common point.
 +
        ii. The perpendicular bisectors of the sides of <math>\mathcal{P}</math> are all distinct.
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        iii. If the perpendicular bisectors of the sides of <math>\mathcal{P}</math> all share at least one common point then <math>\mathcal{P}</math> is a square.
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        iv. If the perpendicular bisectors of the sides of <math>\mathcal{P}</math> are all distinct then these bisectors form a parallelogram.
 +
 
 +
<math>\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 2\qquad\textbf{(D) } 3\qquad\textbf{(E) } 4\qquad</math>
  
 
[[2021 CMC 12A Problems/Problem 6|Solution]]
 
[[2021 CMC 12A Problems/Problem 6|Solution]]
 
==Problem 7==
 
==Problem 7==
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Hold your horses! The test hasn't finished yet.
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It is known that every positive integer can be represented as the sum of at most <math>4</math> squares. What is the sum of the <math>2</math> smallest integers which cannot be represented as the sum of fewer than <math>4</math> squares?
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 +
<math>\textbf{(A) } 22\qquad\textbf{(B) } 23\qquad\textbf{(C) } 24\qquad\textbf{(D) } 25\qquad\textbf{(E) } 26\qquad</math>
  
 
[[2021 CMC 12A Problems/Problem 7|Solution]]
 
[[2021 CMC 12A Problems/Problem 7|Solution]]
 
==Problem 8==
 
==Problem 8==
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Hold your horses! The test hasn't finished yet.
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There are <math>8</math> scoops of ice cream, two of each flavor: vanilla, strawberry, cherry, and mint. If scoops of the same flavor are not distinguishable, how many ways are there to distribute one scoop of ice cream to each of <math>5</math> different people?
 +
 
 +
<math>\textbf{(A) } 420\qquad\textbf{(B) } 450\qquad\textbf{(C) } 600\qquad\textbf{(D) } 1620\qquad\textbf{(E) } 2520\qquad</math>
  
 
[[2021 CMC 12A Problems/Problem 8|Solution]]
 
[[2021 CMC 12A Problems/Problem 8|Solution]]
 
==Problem 9==
 
==Problem 9==
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Hold your horses! The test hasn't finished yet.
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Suppose points <math>A, B, C, D, E,</math> and <math>F</math> lie on a line such that <math>AB=1, BC=4, CD=9, DE=16, EF=25,</math> and <math>FA=23</math>. What is <math>CF</math>?
 +
 
 +
<math>\textbf{(A) } 14\qquad\textbf{(B) } 18\qquad\textbf{(C) } 21\qquad\textbf{(D) } 25\qquad\textbf{(E) } 26\qquad</math>
  
 
[[2021 CMC 12A Problems/Problem 9|Solution]]
 
[[2021 CMC 12A Problems/Problem 9|Solution]]
 +
==Problem 10==
 +
A soccer league consists of <math>20</math> teams and every pair of teams play each other once. If the game is a draw then each team receives one point, otherwise the winner receives <math>3</math> points while the loser receives <math>0</math> points. If the total number of points scored by all the teams was <math>500,</math> how many games ended in a draw?
  
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==Problem 10==
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<math>\textbf{(A) } 70\qquad\textbf{(B) } 75\qquad\textbf{(C) } 100\qquad\textbf{(D) } 115\qquad\textbf{(E) } 120\qquad</math>
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Hold your horses! The test hasn't finished yet.
 
  
 
[[2021 CMC 12A Problems/Problem 10|Solution]]
 
[[2021 CMC 12A Problems/Problem 10|Solution]]
 
==Problem 11==
 
==Problem 11==
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[[2021 CMC 12A Problems/Problem 11|Solution]]
 
[[2021 CMC 12A Problems/Problem 11|Solution]]
 
==Problem 12==
 
==Problem 12==
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[[2021 CMC 12A Problems/Problem 12|Solution]]
 
[[2021 CMC 12A Problems/Problem 12|Solution]]
 
==Problem 13==
 
==Problem 13==
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[[2021 CMC 12A Problems/Problem 13|Solution]]
 
[[2021 CMC 12A Problems/Problem 13|Solution]]
 
==Problem 14==
 
==Problem 14==
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[[2021 CMC 12A Problems/Problem 14|Solution]]
 
[[2021 CMC 12A Problems/Problem 14|Solution]]
 
==Problem 15==
 
==Problem 15==
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[[2021 CMC 12A Problems/Problem 15|Solution]]
 
[[2021 CMC 12A Problems/Problem 15|Solution]]
 
==Problem 16==
 
==Problem 16==
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[[2021 CMC 12A Problems/Problem 16|Solution]]
 
[[2021 CMC 12A Problems/Problem 16|Solution]]
 
==Problem 17==
 
==Problem 17==
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[[2021 CMC 12A Problems/Problem 17|Solution]]
 
[[2021 CMC 12A Problems/Problem 17|Solution]]
 
==Problem 18==
 
==Problem 18==
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[[2021 CMC 12A Problems/Problem 18|Solution]]
 
[[2021 CMC 12A Problems/Problem 18|Solution]]
 
==Problem 19==
 
==Problem 19==
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[[2021 CMC 12A Problems/Problem 19|Solution]]
 
[[2021 CMC 12A Problems/Problem 19|Solution]]
 
==Problem 20==
 
==Problem 20==
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[[2021 CMC 12A Problems/Problem 20|Solution]]
 
[[2021 CMC 12A Problems/Problem 20|Solution]]
 
==Problem 21==
 
==Problem 21==
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[[2021 CMC 12A Problems/Problem 21|Solution]]
 
[[2021 CMC 12A Problems/Problem 21|Solution]]
 
==Problem 22==
 
==Problem 22==
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[[2021 CMC 12A Problems/Problem 22|Solution]]
 
[[2021 CMC 12A Problems/Problem 22|Solution]]
 
==Problem 23==
 
==Problem 23==
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[[2021 CMC 12A Problems/Problem 23|Solution]]
 
[[2021 CMC 12A Problems/Problem 23|Solution]]
 
==Problem 24==
 
==Problem 24==
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[[2021 CMC 12A Problems/Problem 24|Solution]]
 
[[2021 CMC 12A Problems/Problem 24|Solution]]
 
==Problem 25==
 
==Problem 25==
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[[2021 CMC 12A Problems/Problem 25|Solution]]
 
[[2021 CMC 12A Problems/Problem 25|Solution]]
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{{CMC12 box|year=2021|before=[[2020 CMC 12B Problems]]|after=[[2021 CMC 12B Problems]]}}
  
 
[[Category: AMC 12 Problems]]
 
[[Category: AMC 12 Problems]]

Latest revision as of 08:35, 8 August 2023

2021 CMC 12A (Answer Key)
Printable version: Wiki | AoPS Resources

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

What is the value of \[\frac{21^2+\tfrac{21}{20}}{20^2+\tfrac{20}{21}}?\]

$\textbf{(A) }\tfrac{400}{441}\qquad\textbf{(B) }\tfrac{20}{21}\qquad\textbf{(C) } 1\qquad\textbf{(D) }\tfrac{21}{20}\qquad\textbf{(E) }\tfrac{441}{400}\qquad$

Solution

Problem 2

Two circles of equal radius $r$ have an overlap area of $7\pi$ and the total area covered by the circles is $25\pi$. What is the value of $r$?

$\textbf{(A) } 2\sqrt{2}\qquad\textbf{(B) } 2\sqrt{3}\qquad\textbf{(C) } 4\qquad\textbf{(D) } 5\qquad\textbf{(E) } 4\sqrt{2}\qquad$

Solution

Problem 3

A pyramid whose base is a regular $n$-gon has the same number of edges as a prism whose base is a regular $m$-gon. What is the smallest possible value of $n$?

$\textbf{(A) } 3\qquad\textbf{(B) } 4\qquad\textbf{(C) } 6\qquad\textbf{(D) } 9\qquad\textbf{(E) } 12\qquad$

Solution

Problem 4

There exists a positive integer $N$ such that \[\frac{\tfrac{1}{999}+\tfrac{1}{1001}}{2}=\frac{1}{1000}+\frac{1}{N}\] What is the sum of the digits of $N$?

$\textbf{(A) } 36\qquad\textbf{(B) } 45\qquad\textbf{(C) } 54\qquad\textbf{(D) } 63\qquad\textbf{(E) } 72\qquad$

Solution

Problem 5

For positive integers $m>1$ and $n>1,$ the sum of the first $m$ multiples of $n$ is $2020$. Compute $m+n$.

$\textbf{(A) } 206\qquad\textbf{(B) } 208\qquad\textbf{(C) } 210\qquad\textbf{(D) } 212\qquad\textbf{(E) } 214\qquad$

Solution

Problem 6

How many of the following statements are true for every parallelogram $\mathcal{P}$? 

       i. The perpendicular bisectors of the sides of $\mathcal{P}$ all share at least one common point.
       ii. The perpendicular bisectors of the sides of $\mathcal{P}$ are all distinct.
       iii. If the perpendicular bisectors of the sides of $\mathcal{P}$ all share at least one common point then $\mathcal{P}$ is a square.
       iv. If the perpendicular bisectors of the sides of $\mathcal{P}$ are all distinct then these bisectors form a parallelogram.

$\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 2\qquad\textbf{(D) } 3\qquad\textbf{(E) } 4\qquad$

Solution

Problem 7

It is known that every positive integer can be represented as the sum of at most $4$ squares. What is the sum of the $2$ smallest integers which cannot be represented as the sum of fewer than $4$ squares?

$\textbf{(A) } 22\qquad\textbf{(B) } 23\qquad\textbf{(C) } 24\qquad\textbf{(D) } 25\qquad\textbf{(E) } 26\qquad$

Solution

Problem 8

There are $8$ scoops of ice cream, two of each flavor: vanilla, strawberry, cherry, and mint. If scoops of the same flavor are not distinguishable, how many ways are there to distribute one scoop of ice cream to each of $5$ different people?

$\textbf{(A) } 420\qquad\textbf{(B) } 450\qquad\textbf{(C) } 600\qquad\textbf{(D) } 1620\qquad\textbf{(E) } 2520\qquad$

Solution

Problem 9

Suppose points $A, B, C, D, E,$ and $F$ lie on a line such that $AB=1, BC=4, CD=9, DE=16, EF=25,$ and $FA=23$. What is $CF$?

$\textbf{(A) } 14\qquad\textbf{(B) } 18\qquad\textbf{(C) } 21\qquad\textbf{(D) } 25\qquad\textbf{(E) } 26\qquad$

Solution

Problem 10

A soccer league consists of $20$ teams and every pair of teams play each other once. If the game is a draw then each team receives one point, otherwise the winner receives $3$ points while the loser receives $0$ points. If the total number of points scored by all the teams was $500,$ how many games ended in a draw?

$\textbf{(A) } 70\qquad\textbf{(B) } 75\qquad\textbf{(C) } 100\qquad\textbf{(D) } 115\qquad\textbf{(E) } 120\qquad$

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

Problem 16

Solution

Problem 17

Solution

Problem 18

Solution

Problem 19

Solution

Problem 20

Solution

Problem 21

Solution

Problem 22

Solution

Problem 23

Solution

Problem 24

Solution

Problem 25

Solution

2021 CMC 12 (Problems • Answer Key • Resources)
Preceded by
2020 CMC 12B Problems 		Followed by
2021 CMC 12B 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 CMC 12 Problems and Solutions
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