2022 AMC 12A Problems
2022 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
The sum of three numbers is The first number is
times the third number, and the third number is
less than the second number. What is the absolute value of the difference between the first and second numbers?
Problem 3
These problems will be posted once the 2022 AMC 12A is released.
Problem 4
The least common multiple of a positive divisor and
is
, and the greatest common divisor of
and
is
. What is the sum of the digits of
?
Problem 5
The between points
and
in the coordinate plane is given by
. For how many points
with integer coordinates is the taxicab distance between
and the origin less than or equal to
?
Problem 6
A data set consists of not distinct) positive integers:
,
,
,
,
, and
. The
average (arithmetic mean) of the
numbers equals a value in the data set. What is
the sum of all positive values of
?
Problem 7
A rectangle is partitioned into regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?
Problem 8
The infinite product
evaluates to a real number. What is that number?
Problem 9
On Halloween children walked into the principal's office asking for candy. They
can be classified into three types: Some always lie; some always tell the truth; and
some alternately lie and tell the truth. The alternaters arbitrarily choose their first
response, either a lie or the truth, but each subsequent statement has the opposite
truth value from its predecessor. The principal asked everyone the same three
questions in this order.
"Are you a truth-teller?" The principal gave a piece of candy to each of the
children who answered yes.
"Are you an alternater?" The principal gave a piece of candy to each of the
children who answered yes.
"Are you a liar?" The principal gave a piece of candy to each of the children who
answered yes.
How many pieces of candy in all did the principal give to the children who always tell the truth?
Problem 10
How many ways are there to split the integers through
into
pairs such that in each pair, the greater number is at least
times the lesser number?
Problem 11
What is the product of all real numbers such that the distance on the number line between
and
is twice the distance on the number line between
and
?
Problem 12
Let be the midpoint of
in regular tetrahedron
. What is
?
Problem 13
Let be the region in the complex plane consisting of all complex numbers
that can be written as the sum of complex numbers
and
, where
lies on the segment with endpoints
and
, and
has magnitude at most
. What integer is closest to the area of
?
Problem 14
These problems will be posted once the 2022 AMC 12A is released.
Problem 15
These problems will be posted once the 2022 AMC 12A is released.
Problem 16
These problems will be posted once the 2022 AMC 12A is released.
Problem 17
These problems will be posted once the 2022 AMC 12A is released.
Problem 18
These problems will be posted once the 2022 AMC 12A is released.
Problem 19
These problems will be posted once the 2022 AMC 12A is released.
Problem 20
These problems will be posted once the 2022 AMC 12A is released.
Problem 21
These problems will be posted once the 2022 AMC 12A is released.
Problem 22
These problems will be posted once the 2022 AMC 12A is released.
Problem 23
Let and
be the unique relatively prime positive integers such that
Let
denote the least common multiple of the numbers
. For how many integers with
is
?
Problem 24
How many strings of length formed from the digits
,
,
,
,
are there such that for each
, at least
of the digits are less than
? (For example,
satisfies this condition
because it contains at least
digit less than
, at least
digits less than
, at least
digits less
than
, and at least
digits less than
. The string
does not satisfy the condition because it
does not contain at least
digits less than
.)
Problem 25
A circle with integer radius is centered at
. Distinct line segments of length
connect points
to
for
and are tangent to the circle, where
,
, and
are all positive integers and
. What is the ratio
for the least possible value of
?
See also
2022 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by 2021 AMC 12B Problems |
Followed by 2022 AMC 12B Problems |
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All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.