Difference between revisions of "2021 CMC 12A Problems"
Sugar rush (talk | contribs) |
(→Problem 4) |
||
Line 19: | Line 19: | ||
[[2021 CMC 12A Problems/Problem 3|Solution]] | [[2021 CMC 12A Problems/Problem 3|Solution]] | ||
==Problem 4== | ==Problem 4== | ||
− | 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 09:35, 8 August 2023
2021 CMC 12A (Answer Key) Printable version: | AoPS Resources | ||
Instructions
| ||
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 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
Problem 1
What is the value of
Problem 2
Two circles of equal radius have an overlap area of
and the total area covered by the circles is
. What is the value of
?
Problem 3
A pyramid whose base is a regular -gon has the same number of edges as a prism whose base is a regular
-gon. What is the smallest possible value of
?
Problem 4
There exists a positive integer such that
What is the sum of the digits of
?
Problem 5
For positive integers and
the sum of the first
multiples of
is
. Compute
.
Problem 6
How many of the following statements are true for every parallelogram ?
i. The perpendicular bisectors of the sides ofall share at least one common point. ii. The perpendicular bisectors of the sides of
are all distinct. iii. If the perpendicular bisectors of the sides of
all share at least one common point then
is a square. iv. If the perpendicular bisectors of the sides of
are all distinct then these bisectors form a parallelogram.
Problem 7
It is known that every positive integer can be represented as the sum of at most squares. What is the sum of the
smallest integers which cannot be represented as the sum of fewer than
squares?
Problem 8
There are 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
different people?
Problem 9
Suppose points and
lie on a line such that
and
. What is
?
Problem 10
A soccer league consists of 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
points while the loser receives
points. If the total number of points scored by all the teams was
how many games ended in a draw?
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
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 |