2021 CMC 12A Problems
2021 CMC 12A (Answer Key) Printable version: | AoPS Resources | ||
Instructions
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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 of all 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 |
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All CMC 12 Problems and Solutions |