2023 AMC 10A Problems
2023 AMC 10A (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
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 |
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
Cities and are miles apart. Alicia lives in and Beth lives in . Alicia bikes towards at miles per hour. Leaving at the same time, Beth bikes toward at miles per hour. How many miles from City will they be when they meet?
Problem 2
The weight of of a large pizza together with cups of orange slices is the same weight of of a large pizza together with cups of orange slices. A cup of orange slices weigh of a pound. What is the weight, in pounds, of a large pizza?
Problem 3
How many positive perfect squares less than are divisible by ?
Problem 4
A quadrilateral has all integer side lengths, a perimeter of , and one side of length . What is the greatest possible length of one side of this quadrilateral?
Problem 5
How many digits are in the base-ten representation of ?
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 . What is the value of the cube?
Problem 7
Janet rolls a standard -sided die times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal
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 degrees Fahrenheit, which is degrees on the Breadus scale. Bread is baked at degrees Fahrenheit, which is degrees on the Breadus scale. Bread is done when its internal temperature is degrees Fahrenheit. What is this, in degrees, on the Breadus scale?
Problem 9
A digital display shows the current date as an -digit integer consisting of a -digit year, followed by a -digit month, followed by a -digit date within the month. For example, Arbor Day this year is displayed as For how many dates in does each digit appear an even number of times in the -digital display for that date?
Problem 10
Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an on the next quiz, her mean will increase by . If she scores an on each of the next three quizzes, her mean will increase by . What is the mean of her quiz scores currently?
Problem 11
A square of area is inscribed in a square of area , creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?
Problem 12
How many three-digit positive integers satisfy the following properties?
- The number is divisible by .
- The number formed by reversing the digits of is divisible by .
Problem 13
Abdul and Chiang are standing 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 . What is the square of the distance (in feet) between Abdul and Bharat?
Problem 14
A number is chosen at random from among the first 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 ?
Problem 15
An even number of circles are nested, starting with a radius of and increasing by each time, all sharing a common point. The region between every other circle is shaded, starting with the region inside the circle of radius but outside the circle of radius An example showing circles is displayed below. What is the least number of circles needed to make the total shaded area at least ?
Problem 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 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?
Problem 17
Let be a rectangle with and . Point and lie on and respectively so that all sides of and have integer lengths. What is the perimeter of ?
Problem 18
A rhombic dodecahedron is a solid with congruent rhombus faces. At every vertex, or edges meet, depending on the vertex. How many vertices have exactly edges meet?
Problem 19
The line segment formed by and is rotated to the line segment formed by and about the point . What is ?
Problem 20
Each square in a grid of squares is colored red, white, blue, or green so that every square contains one square of each color. One such coloring is shown on the right below. How many different colorings are possible?
Problem 21
Let be the unique polynomial of minimal degree with the following properties:
- has a leading coefficient ,
- is a root of ,
- is a root of ,
- is a root of , and
- is a root of .
The roots of are integers, with one exception. The root that is not an integer can be written as , where and are relatively prime integers. What is ?
Problem 22
Circle and each have radius , and the distance between their centers is . Circle is the largest circle internally tangent to both and . Circle is internally tangent to both and and externally tangent to . What is the radius of ?
Problem 23
If the positive integer has positive integer divisors and with , then and are said to be divisors of . Suppose that is a positive integer that has one complementary pair of divisors that differ by and another pair of complementary divisors that differ by . What is the sum of the digits of ?
Problem 24
Six regular hexagonal blocks of side length 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 unit. What is the area of the region inside the frame not occupied by the blocks?
Problem 25
If and are vertices of a polyhedron, define the distance to be the minimum number of edges of the polyhedron one must traverse in order to connect and . For example, is an edge of the polyhedron, then , but if and are edges and is not an edge, then . Let , , and be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of equilateral triangles). What is the probability that ?
See also
2023 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by 2022 AMC 10B Problems |
Followed by 2023 AMC 10B 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 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.