AMC 10A 2023

25 Problems • Answer Key • 2 of 25 Solutions Available

2023 AMC 10A Practice Test

Take the complete 2023 AMC 10A 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 2023 (Problem 4)

A quadrilateral has all integer side lengths, a perimeter of

26

, and one side of length

4

. What is the greatest possible length of one side of this quadrilateral?

\text{(A)}\;9

\text{(B)}\;10

\text{(C)}\;11

\text{(D)}\;12

\text{(E)}\;13

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 2023 (Problem 6)

An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is

21

. What is the value of the cube?

\text{(A)}\;42

\text{(B)}\;63

\text{(C)}\;84

\text{(D)}\;126

\text{(E)}\;252

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 10A 2023 (Problem 8)

Barb the baker has developed a new temperature scale for her bakery called the Breadus scale, which is a linear function of the Fahrenheit scale. Bread rises at

110

degrees Fahrenheit, which is

0

degrees on the Breadus scale. Bread is baked at

350

degrees Fahrenheit, which is

100

degrees on the Breadus scale. Bread is done when its internal temperature is

200

degrees Fahrenheit. What is this, in degrees, on the Breadus scale?

\text{(A)}\;33

\text{(B)}\;34.5

\text{(C)}\;36

\text{(D)}\;37.5

\text{(E)}\;39

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)

10.

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)

11.

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 10A 2023 (Problem 12)

12.

How many three-digit positive integers

N

satisfy the following properties?

The number $N$ is divisible by $7$ .
The number formed by reversing the digits of $N$ is divisible by $5$ .

\text{(A)}\;13

\text{(B)}\;14

\text{(C)}\;15

\text{(D)}\;16

\text{(E)}\;17

Full solution coming soon

AMC 10A 2023 (Problem 13)

13.

Abdul and Chiang are standing

48

feet apart in a field. Bharat is standing in the same field as far from Abdul as possible so that the angle formed by his lines of sight to Abdul and Chiang measures

60^\circ

. What is the square of the distance (in feet) between Abdul and Bharat?

\text{(A)}\;1728

\text{(B)}\;2601

\text{(C)}\;3072

\text{(D)}\;4608

\text{(E)}\;6912

Full solution coming soon

AMC 10A 2023 (Problem 14)

14.

A number is chosen at random from among the first

100

positive integers, and a positive integer divisor of that number is then chosen at random. What is the probability that the chosen divisor is divisible by

11

\text{(A)}\;\frac{4}{100}

\text{(B)}\;\frac{9}{200}

\text{(C)}\;\frac{1}{20}

\text{(D)}\;\frac{11}{200}

\text{(E)}\;\frac{3}{50}

Full solution coming soon

AMC 10A 2023 (Problem 15)

15.

An even number of circles are nested, starting with a radius of

1

and increasing by

1

each time, all sharing a common point. The region between every other circle is shaded, starting with the region inside the circle of radius

2

but outside the circle of radius

1

. An example showing

8

circles is displayed below. What is the least number of circles needed to make the total shaded area at least

2023\pi

\text{(A)}\;46

\text{(B)}\;48

\text{(C)}\;56

\text{(D)}\;60

\text{(E)}\;64

Full solution coming soon

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

16.

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 10A 2023 (Problem 17)

17.

Let

ABCD

be a rectangle with

AB = 30

and

BC = 28

. Point

P

and

Q

lie on

\overline{BC}

and

\overline{CD}

respectively so that all sides of

\triangle ABP

\triangle PCQ

, and

\triangle QDA

have integer lengths. What is the perimeter of

\triangle APQ

\text{(A)}\;84

\text{(B)}\;86

\text{(C)}\;88

\text{(D)}\;90

\text{(E)}\;92

Full solution coming soon

AMC 10A 2023 (Problem 18)

18.

A rhombic dodecahedron is a solid with

12

congruent rhombus faces. At every vertex,

3

4

edges meet, depending on the vertex. How many vertices have exactly

3

edges meet?

\text{(A)}\;5

\text{(B)}\;6

\text{(C)}\;7

\text{(D)}\;8

\text{(E)}\;9

Full solution coming soon

AMC 10A 2023 (Problem 19)

19.

The line segment formed by

A(1, 2)

and

B(3, 3)

is rotated to the line segment formed by

A\prime(3, 1)

and

B\prime(4, 3)

about the point

P(r, s)

. What is

|r-s|

\text{(A)}\;\frac14

\text{(B)}\;\frac12

\text{(C)}\;\frac34

\text{(D)}\;\frac23

\text{(E)}\;1

Full solution coming soon

AMC 10A 2023 (Problem 20)

20.

Each square in a

3\times3

grid of squares is colored red, white, blue, or green so that every

2\times2

square contains one square of each color. One such coloring is shown on the right below. How many different colorings are possible?

\text{(A)}\;24

\text{(B)}\;48

\text{(C)}\;60

\text{(D)}\;72

\text{(E)}\;96

Full solution coming soon

AMC 10A 2023 (Problem 21)

21.

Let

P(x)

be the unique polynomial of minimal degree with the following properties:

$P(x)$ has a leading coefficient $1$ .
$1$ is a root of $P(x)-1$ .
$2$ is a root of $P(x-2)$ .
$3$ is a root of $P(3x)$ .
$4$ is a root of $4P(x)$ .

The roots of

P(x)

are integers, with one exception. The root that is not an integer can be written as

\frac{m}{n}

, where

m

and

n

are relatively prime integers. What is

m+n

\text{(A)}\;41

\text{(B)}\;43

\text{(C)}\;45

\text{(D)}\;47

\text{(E)}\;49

Full solution coming soon

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

Plane GeometryHard

22.

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 10A 2023 (Problem 23)

Number TheoryMedium

23.

If the positive integer

c

has positive integer divisors

a

and

b

with

c=ab

, then

a

and

b

are said to be complementary divisors of

c

. Suppose that

N

is a positive integer that has one complementary pair of divisors that differ by

20

and another pair of complementary divisors that differ by

23

. What is the sum of the digits of

N

\text{(A)}\;9

\text{(B)}\;13

\text{(C)}\;15

\text{(D)}\;17

\text{(E)}\;19

AMC 10A 2023 (Problem 24)

24.

Six regular hexagonal blocks of side length

1

unit are arranged inside a regular hexagonal frame. Each block lies along an inside edge of the frame and is aligned with two other blocks, as shown in the figure below. The distance from any corner of the frame to the nearest vertex of a block is

\frac37

unit. What is the area of the region inside the frame not occupied by the blocks?

\text{(A)}\;\frac{13\sqrt3}{3}

\text{(B)}\;\frac{216\sqrt3}{49}

\text{(C)}\;\frac{9\sqrt3}{2}

\text{(D)}\;\frac{14\sqrt3}{3}

\text{(E)}\;\frac{243\sqrt3}{49}

Full solution coming soon

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

25.

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