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Difference between revisions of "2021 CMC 12A Problems"
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[[2021 CMC 12A Problems/Problem 3|Solution]]
 
[[2021 CMC 12A Problems/Problem 3|Solution]]
 
==Problem 4==
 
==Problem 4==
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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}{N}</cmath> What is the sum of the digits of <math>N</math>?
+
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>?
  
 
<math>\textbf{(A) } 36\qquad\textbf{(B) } 45\qquad\textbf{(C) } 54\qquad\textbf{(D) } 63\qquad\textbf{(E) } 72\qquad</math>
 
<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]]
 +
 
==Problem 5==
 
==Problem 5==
 
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>.
 
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>.

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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