2022 AMC 12B Problems
2022 AMC 12B (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
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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
Define to be for all real numbers and What is the value of
Problem 2
In rhombus , point lies on segment so that , , and . What is the area of ? (Note: The figure is not drawn to scale.)
Problem 3
How many of the first ten numbers of the sequence are prime numbers?
Problem 4
For how many values of the constant will the polynomial have two distinct integer roots?
Problem 5
The point is rotated counterclockwise about the point . What are the coordinates of its new position?
Problem 6
Consider the following sets of elements each: How many of these sets contain exactly two multiples of ?
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?
Problem 8
What is the graph of in the coordinate plane?
Problem 9
The sequence is a strictly increasing arithmetic sequence of positive integers such that What is the minimum possible value of ?
Problem 10
Regular hexagon has side length . Let be the midpoint of , and let be the midpoint of . What is the perimeter of ?
Problem 11
Let , where . What is ?
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 ?
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?
Problem 14
The graph of intersects the -axis at points and and the -axis at point . What is ?
Problem 15
One of the following numbers is not divisible by any prime number less than Which is it?
Problem 16
Suppose and are positive real numbers such that What is the greatest possible value of ?
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.
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. 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.)
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
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?
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 ?
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 ?
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
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?
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 12-sided outer nonconvex polygon can be written as , where , , and are integers and is not divisible by the square of any prime. What is ?
See also
2022 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by 2022 AMC 12A Problems |
Followed by 2023 AMC 12A Problems |
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All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.