AMC 12A 2023

25 Problems • Answer Key • 9 of 25 Solutions Available

2023 AMC 12A Practice Test

Take the complete 2023 AMC 12A as a timed practice test. You'll have 75 minutes to solve all 25 questions under realistic AMC conditions. Submit your test to see your score and review the answer key. Detailed solutions are being added progressively.

AMC 10A/12A 2023 (Problem 1/1)

Cities

A

and

B

are

45

miles apart. Alicia lives in

A

and Beth lives in

B

. Alicia bikes towards

B

18

miles per hour. Leaving at the same time, Beth bikes toward

A

12

miles per hour. How many miles from City

A

will they be when they meet?

\text{(A)}\;20

\text{(B)}\;24

\text{(C)}\;25

\text{(D)}\;26

\text{(E)}\;27

Full solution coming soon

AMC 10A/12A 2023 (Problem 2/2)

The weight of

\frac{1}{3}

of a large pizza together with

3 \frac{1}{2}

cups of orange slices is the same weight of

\frac{3}{4}

of a large pizza together with

\frac{1}{2}

cups of orange slices. A cup of orange slices weigh

\frac{1}{4}

of a pound. What is the weight, in pounds, of a large pizza?

\text{(A)}\;1\frac{4}{5}

\text{(B)}\;2

\text{(C)}\;2\frac{2}{5}

\text{(D)}\;3

\text{(E)}\;3\frac{3}{5}

Full solution coming soon

AMC 10A/12A 2023 (Problem 3/3)

How many positive perfect squares less than

2023

are divisible by

5

\text{(A)}\;8

\text{(B)}\;9

\text{(C)}\;10

\text{(D)}\;11

\text{(E)}\;12

Full solution coming soon

AMC 10A/12A 2023 (Problem 5/4)

How many digits are in the base-ten representation of

8^5 \cdot 5^{10} \cdot 15^5

\text{(A)}\;14

\text{(B)}\;15

\text{(C)}\;16

\text{(D)}\;17

\text{(E)}\;18

Full solution coming soon

AMC 10A/12A 2023 (Problem 7/5)

Janet rolls a standard

6

-sided die

4

times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal

3

\text{(A)}\;\frac{2}{9}

\text{(B)}\;\frac{49}{216}

\text{(C)}\;\frac{25}{108}

\text{(D)}\;\frac{17}{72}

\text{(E)}\;\frac{13}{54}

Full solution coming soon

AMC 12A 2023 (Problem 6)

Points

A

and

B

lie on the graph of

y=\log_{2}x

. The midpoint of

\overline{AB}

(6, 2)

. What is the positive difference between the

x

-coordinates of

A

and

B

\text{(A)}\;2\sqrt{11}

\text{(B)}\;4\sqrt{3}

\text{(C)}\;8

\text{(D)}\;4\sqrt{5}

\text{(E)}\;9

Full solution coming soon

AMC 10A/12A 2023 (Problem 9/7)

A digital display shows the current date as an

8

-digit integer consisting of a

4

-digit year, followed by a

2

-digit month, followed by a

2

-digit date within the month. For example, Arbor Day this year is displayed as

20230428

. For how many dates in

2023

does each digit appear an even number of times in the

8

-digit display for that date?

\text{(A)}\;5

\text{(B)}\;6

\text{(C)}\;7

\text{(D)}\;8

\text{(E)}\;9

Full solution coming soon

AMC 10A/12A 2023 (Problem 10/8)

Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an

11

on the next quiz, her mean will increase by

1

. If she scores an

11

on each of the next three quizzes, her mean will increase by

2

. What is the mean of her quiz scores currently?

\text{(A)}\;4

\text{(B)}\;5

\text{(C)}\;6

\text{(D)}\;7

\text{(E)}\;8

Full solution coming soon

AMC 10A/12A 2023 (Problem 11/9)

A square of area

2

is inscribed in a square of area

3

, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?

\text{(A)}\;\frac15

\text{(B)}\;\frac14

\text{(C)}\;2-\sqrt3

\text{(D)}\;\sqrt3-\sqrt2

\text{(E)}\;\sqrt2-1

Full solution coming soon

AMC 12A 2023 (Problem 10)

EquationsEasy

10.

Positive real numbers

x

and

y

satisfy

y^3=x^2

and

(y-x)^2=4y^2

. What is

x+y

\text{(A)}\;12

\text{(B)}\;18

\text{(C)}\;24

\text{(D)}\;36

\text{(E)}\;42

AMC 12A 2023 (Problem 11)

11.

What is the degree measure of the acute angle formed by lines with slopes

2

and

\tfrac{1}{3}

\text{(A)}\;30

\text{(B)}\;37.5

\text{(C)}\;45

\text{(D)}\;52.5

\text{(E)}\;60

Full solution coming soon

AMC 12A 2023 (Problem 12)

Expressions & IdentitiesEasy

12.

What is the value of

2^3-1^3+4^3-3^3+6^3-5^3+\cdots+18^3-17^3

\text{(A)}\;2023

\text{(B)}\;2679

\text{(C)}\;2941

\text{(D)}\;3159

\text{(E)}\;3235

AMC 10A/12A 2023 (Problem 16/13)

13.

In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was

40\%

more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?

\text{(A)}\;15

\text{(B)}\;36

\text{(C)}\;45

\text{(D)}\;48

\text{(E)}\;66

Full solution coming soon

AMC 12A 2023 (Problem 14)

14.

How many complex numbers satisfy the equation

z^{5}=\overline{z}

, where

\overline{z}

is the conjugate of the complex number

z

\text{(A)}\;2

\text{(B)}\;3

\text{(C)}\;5

\text{(D)}\;6

\text{(E)}\;7

Full solution coming soon

AMC 12A 2023 (Problem 15)

15.

Usain is walking for exercise by zigzagging across a

100

-meter by

30

-meter rectangular field, beginning at point

A

and ending on the segment

\overline{BC}

. He wants to increase the distance walked by zigzagging as shown in the figure below

(APQRS)

. What angle

\theta

\angle PAB=\angle QPC=\angle RQB=\cdots

will produce a length that is

120

meters? (This figure is not drawn to scale. Do not assume that the zigzag path has exactly four segments as shown; there could be more or fewer.)

Zigzag path across a 100-meter by 30-meter rectangular field

\text{(A)}\;\arccos\frac{5}{6}

\text{(B)}\;\arccos\frac{4}{5}

\text{(C)}\;\arccos\frac{3}{10}

\text{(D)}\;\arcsin\frac{4}{5}

\text{(E)}\;\arcsin\frac{5}{6}

Full solution coming soon

AMC 12A 2023 (Problem 16)

16.

Consider the set of complex numbers

z

satisfying

|1+z+z^{2}|=4

. The maximum value of the imaginary part of

z

can be written in the form

\tfrac{\sqrt{m}}{n}

, where

m

and

n

are relatively prime positive integers. What is

m+n

\text{(A)}\;20

\text{(B)}\;21

\text{(C)}\;22

\text{(D)}\;23

\text{(E)}\;24

Full solution coming soon

AMC 12A 2023 (Problem 17)

ProbabilityMedium

17.

Flora the frog starts at

0

on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance

m

with probability

\frac{1}{2^m}

. What is the probability that Flora will eventually land at

10

\text{(A)}\;\frac{5}{512}

\text{(B)}\;\frac{45}{1024}

\text{(C)}\;\frac{127}{1024}

\text{(D)}\;\frac{511}{1024}

\text{(E)}\;\frac{1}{2}

AMC 10A/12A 2023 (Problem 22/18)

Plane GeometryHard

18.

Circle

C_1

and

C_2

each have radius

1

, and the distance between their centers is

\frac12

. Circle

C_3

is the largest circle internally tangent to both

C_1

and

C_2

. Circle

C_4

is internally tangent to both

C_1

and

C_2

and externally tangent to

C_3

. What is the radius of

C_4

\text{(A)}\;\frac{1}{14}

\text{(B)}\;\frac{1}{12}

\text{(C)}\;\frac{1}{10}

\text{(D)}\;\frac{3}{28}

\text{(E)}\;\frac{1}{9}

AMC 12A 2023 (Problem 19)

FunctionsHard

19.

What is the product of all the solutions to the equation

\log_{7x}2023\cdot\log_{289x}2023=\log_{2023x}2023?

\text{(A)}\;(\log_{2023}7\cdot\log_{2023}289)^2

\text{(B)}\;\log_{2023}7\cdot\log_{2023}289

\text{(C)}\;1

\text{(D)}\;\log_72023\cdot\log_{289}2023

\text{(E)}\;(\log_72023\cdot\log_{289}2023)^2

AMC 12A 2023 (Problem 20)

Sequences & SeriesHard

20.

Rows

1,2,3,4,

and

5

of a triangular array of integers are shown below:

\begin{array}{ccccccccc}&&&&1&&&&\\[6pt]&&&1&&1&&&\\[6pt]&&1&&3&&1&&\\[6pt]&1&&5&&5&&1&\\[6pt]1&&7&&11&&7&&1\end{array}

Each row after the first row is formed by placing a

1

at each end of the row, and each interior entry is

1

greater than the sum of the two numbers diagonally above it in the previous row. What is the units digit of the sum of the

2023

numbers in the

2023

rd row?

\text{(A)}\;1

\text{(B)}\;3

\text{(C)}\;5

\text{(D)}\;7

\text{(E)}\;9

AMC 10A/12A 2023 (Problem 25/21)

21.

A

and

B

are vertices of a polyhedron, define the distance

d(A,B)

to be the minimum number of edges of the polyhedron one must traverse in order to connect

A

and

B

. For example, if

\overline{AB}

is an edge of the polyhedron, then

d(A,B)=1

, but if

\overline{AC}

and

\overline{CB}

are edges and

\overline{AB}

is not an edge, then

d(A,B)=2

. Let

Q

R

, and

S

be randomly chosen distinct vertices of a regular icosahedron (a regular polyhedron made up of

20

equilateral triangles). What is the probability that

d(Q,R)>d(R,S)

\text{(A)}\;\frac7{22}

\text{(B)}\;\frac13

\text{(C)}\;\frac38

\text{(D)}\;\frac5{12}

\text{(E)}\;\frac12

Full solution coming soon

AMC 12A 2023 (Problem 22)

Number TheoryMedium

22.

Let

f

be the unique function defined on the positive integers such that

\sum_{d\mid n} d\cdot f\left(\frac{n}{d}\right)=1

for all positive integers

n

, where the sum is taken over all positive divisors of

n

. What is

f(2023)

\text{(A)}\;-1536

\text{(B)}\;96

\text{(C)}\;108

\text{(D)}\;116

\text{(E)}\;144

AMC 12A 2023 (Problem 23)

InequalitiesHard

23.

How many ordered pairs of positive real numbers

(a,b)

satisfy the equation

(1+2a)(2+2b)(2a+b)=32ab

\text{(A)}\;0

\text{(B)}\;1

\text{(C)}\;2

\text{(D)}\;3

\text{(E)}\;\text{an infinite number}

AMC 12A 2023 (Problem 24)

CombinatoricsHard

24.

Let

K

be the number of sequences

A_1,A_2,\ldots,A_n

such that

n

is a positive integer less than or equal to

10

, each

A_i

is a subset of

\{1,2,3,\ldots,10\}

, and

A_{i-1}

is a subset of

A_i

for each

i

between

2

and

n

, inclusive. For example,

\{\},\{5,7\},\{2,5,7\},\{2,5,7\},\{2,5,6,7,9\}

is one such sequence, with

n=5

. What is the remainder when

K

is divided by

10

\text{(A)}\;1

\text{(B)}\;3

\text{(C)}\;5

\text{(D)}\;7

\text{(E)}\;9

AMC 12A 2023 (Problem 25)

25.

There is a unique sequence of integers

a_1,a_2,\ldots,a_{2023}

such that

\tan(2023x)=\dfrac{a_1\tan x+a_3\tan^3 x+\cdots+a_{2023}\tan^{2023}x}{1+a_2\tan^2 x+a_4\tan^4 x+\cdots+a_{2022}\tan^{2022}x}

whenever

\tan(2023x)

is defined. What is

a_{2023}

\text{(A)}\;-2023

\text{(B)}\;-2022

\text{(C)}\;-1

\text{(D)}\;1

\text{(E)}\;2023

Full solution coming soon