¶ 2022 AMC 12B Problems and Solutions
Problem Set Workbook
Access the downloadable workbook for 2022 AMC 12B problems here.
Discussion Forum
Engage in discussion about the 2022 AMC 12B math contest by visiting Random Math AMC 12B 2022 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2022 AMC 12B problems, please refer below:
Problem 1: Define to be for all real numbers and . What is the value of
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 2: In rhombus , point lies on segment so that , and . What is the area of ? (Note: The figure is not drawn to scale.)
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 3: How many of the first ten numbers of the sequence are prime numbers?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 4: For how many values of the constant will the polynomial have two distinct integer roots?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 5: The point is rotated counterclockwise about the point . What are the coordinates of its new position?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 6: Consider the following sets of elements each:
How many of these sets contain exactly two multiples of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 7: Camila writes down five positive integers. The unique mode of these integers is greater than their median, and the median is greater than their arithmetic mean. What is the least possible value for the mode?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 8: What is the graph of in the coordinate plane?
Answer Choices:
A. two intersecting parabolas
B. two nonintersecting parabolas
C. two intersecting circles
D. a circle and a hyperbola
E. a circle and two parabolas
Solution:
Problem 9: The sequence is a strictly increasing arithmetic sequence of positive integers such that
What is the minimum possible value of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 10: Regular hexagon has side length . Let be the midpoint of , and let be the midpoint of . What is the perimeter of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 11: Let
where . What is ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 12: Kayla rolls four fair -sided dice. What is the probability that at least one of the numbers Kayla rolls is greater than and at least two of the numbers she rolls are greater than ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 13: The diagram below shows a rectangle with side lengths and and a square with side length . Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 14: The graph of intersects the -axis at points and and the -axis at point . What is ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 15: One of the following numbers is not divisible by any prime number less than . Which is it?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 16: Suppose and are positive real numbers such that
What is the greatest possible value of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 17: How many arrays whose entries are s and s are there such that the row sums (the sum of the entries in each row) are , and , in some order, and the column sums (the sum of the entries in each column) are also , and , in some order? For example, the array
satisfies the condition.
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 18: Each square in a grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules:
Any filled square with two or three filled neighbors remains filled. Any empty square with exactly three filled neighbors becomes a filled square. All other squares remain empty or become empty. A sample transformation is shown in the figure below.
.jpg)
.jpg)
Initial Transformed
Suppose the grid has a border of empty squares surrounding a subgrid. How many initial configurations will lead to a transformed grid consisting of a single filled square in the center after a single transformation? (Rotations and reflections of the same configuration are considered different.)
.jpg)
Initial Transformed
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 19: In medians and intersect at and is equilateral. Then can be written as , where and are relatively prime positive integers and is a positive integer not divisible by the square of any prime. What is ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 20: Let be a polynomial with rational coefficients such that when is divided by the polynomial , the remainder is , and when is divided by the polynomial , the remainder is . There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 21: Let be the set of circles in the coordinate plane that are tangent to each of the three circles with equations , and . What is the sum of the areas of all circles in ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 22: Ant Amelia starts on the number line at and crawls in the following manner. For , Amelia chooses a time duration and an increment independently and uniformly at random from the interval . During the th step of the process, Amelia moves units in the positive direction, using up minutes. If the total elapsed time has exceeded minute during the th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most steps in all. What is the probability that Amelia's position when she stops will be greater than ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 23: Let be a sequence of numbers, where each is either or . For each positive integer , define
Suppose for all . What is the value of the sum
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 24: The figure below depicts a regular -gon inscribed in a unit circle.
What is the sum of the th powers of the lengths of all of its edges and diagonals?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 25: Four regular hexagons surround a square with a side length , each one sharing an edge with the square, as shown in the figure below. The area of the resulting -sided outer nonconvex polygon can be written as , where , and are integers and is not divisible by the square of any prime. What is ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions