¶ 2018 AMC 10A Problems and Solutions
Problem Set Workbook
Access the downloadable workbook for 2018 AMC 10A problems here.
Discussion Forum
Engage in discussion about the 2018 AMC 10A math contest by visiting Random Math AMC 10A 2018 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2018 AMC 10A problems, please refer below:
Problem 1: What is the value of
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 2: Liliane has more soda than Jacqueline, and Alice has more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alice have?
Answer Choices:
A. Liliane has more soda than Alice.
B. Liliane has more soda than Alice.
C. Liliane has more soda than Alice.
D. Liliane has more soda than Alice.
E. Liliane has more soda than Alice.
Solution:
Problem 3: A unit of blood expires after seconds. Yasin donates a unit of blood at noon on January . On what day does his unit of blood expire?
Answer Choices:
A. January
B. January
C. January
D. February
E. February
Solution:
Problem 4: How many ways can a student schedule mathematics courses - algebra, geometry, and number theory - in a -period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other periods is of no concern here.)
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 5: Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, We are at least miles away, Bob replied, We are at most miles away.Charlie then remarked, Actually the nearest town is at most miles away. It turned out that none of the three statements was true. Let be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 6: Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of , and the score increases by for each like vote and decreases by for each dislike vote. At one point Sangho saw that his video had a score of , and that of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 7: For how many (not necessarily positive) integer values of is the value of an integer?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 8: Joe has a collection of coins, consisting of -cent coins, -cent coins, and -cent coins. He has more -cent coins than -cent coins, and the total value of his collection is cents. How many more -cent coins does Joe have than -cent coins?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 9: All of the triangles in the diagram below are similar to isosceles triangle , in which . Each of the smallest triangles has area , and has area . What is the area of trapezoid ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 10: Suppose that real number satisfies
What is the value of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 11: When fair standard -sided dice are thrown, the probability that the sum of the numbers on the top faces is can be written as
where is a positive integer. What is ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 12: How many ordered pairs of real numbers satisfy the following system of equations?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 13: A paper triangle with sides of lengths , and inches, as shown, is folded so that point falls on point . What is the length in inches of the crease?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 14: What is the greatest integer less than or equal to
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 15: Two circles of radius are externally tangent to each other and are internally tangent to a circle of radius at points and , as shown in the diagram. The distance can be written in the form , where and are relatively prime positive integers. What is ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 16: Right triangle has leg lengths and . Including and , how many line segments with integer length can be drawn from vertex to a point on hypotenuse ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 17: Let be a set of integers taken from with the property that if and are elements of with , then is not a multiple of . What is the least possible value of an element of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 18: How many nonnegative integers can be written in the form
where for ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 19: A number is randomly selected from the set , and a number is randomly selected from . What is the probability that has a units digit of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 20: A scanning code consists of a grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of squares. A scanning code is called symmetric if its look does not change when the entire square is rotated by a multiple of counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 21: Which of the following describes the set of values of for which the curves and in the real -plane intersect at exactly points?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 22: Let , and be positive integers such that , , and . Which of the following must be a divisor of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 23: Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths of and units. In the corner where those sides meet at a right angle, he leaves a small unplanted square so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from to the hypotenuse is units. What fraction of the field is planted?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 24: Triangle with and has area . Let be the midpoint of , and let be the midpoint of . The angle bisector of intersects and at and , respectively. What is the area of quadrilateral ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 25: For a positive integer and nonzero digits , and , let be the -digit integer each of whose digits is equal to ; let be the -digit integer each of whose digits is equal to ; and let be the -digit (not -digit) integer each of whose digits is equal to . What is the greatest possible value of for which there are at least two values of such that
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions