2021 AMC 12A Problems
2021 AMC 12A (Answer Key) Printable versions: • AoPS Resources • PDF | ||
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
- 26 See also
Problem 1
What is the value of
Problem 2
Under what conditions is true, where
and
are real numbers?
It is never true.
It is true if and only if
.
It is true if and only if
.
It is true if and only if
and
.
It is always true.
Problem 3
The sum of two natural numbers is . One of the two numbers is divisible by
. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?
Problem 4
Tom has a collection of snakes,
of which are purple and
of which are happy. He observes that
all of his happy snakes can add,
none of his purple snakes can subtract, and
all of his snakes that can't subtract also can't add.
Which of these conclusions can be drawn about Tom's snakes?
Purple snakes can add.
Purple snakes are happy.
Snakes that can add are purple.
Happy snakes are not purple.
Happy snakes can't subtract.
Problem 5
When a student multiplied the number by the repeating decimal
where
and
are digits, he did not notice the notation and just multiplied
times
. Later he found that his answer is
less than the correct answer. What is the
-digit number
Problem 6
A deck of cards has only red cards and black cards. The probability of a randomly chosen card being red is . When
black cards are added to the deck, the probability of choosing red becomes
. How many cards were in the deck originally?
Problem 7
What is the least possible value of for all real numbers
and
Problem 8
A sequence of numbers is defined by and
for
. What are the parities (evenness or oddness) of the triple of numbers
, where
denotes even and
denotes odd?
Problem 9
Which of the following is equilvalent to
Problem 10
These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
Problem 11
A laser is placed at the point . The laser bean travels in a straight line. Larry wants the beam to hit and bounce off the
-axis, then hit and bounce off the
-axis, then hit the point
. What is the total distance the beam will travel along this path?
Problem 12
All the roots of the polynomial are positive integers, possibly repeated. What is the value of
?
Problem 13
Of the following complex numbers , which one has the property that
has the greatest real part?
Problem 14
What is the value of
Problem 15
A choir director must select a group of singers from among his tenors and
basses. The only
requirements are that the difference between the numbers of tenors and basses must be a multiple
of
, and the group must have at least one singer. Let
be the number of groups that could be
selected. What is the remainder when
is divided by
?
Problem 16
In the following list of numbers, the integer appears
times in the list for
.
What is the median of the numbers in this list?
Problem 17
Trapezoid has
, and
. Let
be the intersection of the diagonals
and
, and let
be the midpoint of
. Given that
, the length of
can be written in the form
, where
and
are positive integers and
is not divisible by the quare of any prime. What is
?
Problem 18
Let be a function defined on the set of positive rational numbers with the property that
for all positive rational numbers
and
. Suppose that
also has the property that
for every prime number
. For which of the following numbers
is
?
Problem 19
These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
Problem 20
These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
Problem 21
These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
Problem 22
These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
Problem 23
These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
Problem 24
These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
Problem 25
These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
See also
2021 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by 2020 AMC 12B Problems |
Followed by 2021 AMC 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 AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.