2022 AMC 10B Problems
2022 AMC 10B (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 three-digit positive integers have an odd number of even digits?
Problem 4
A donkey suffers an attack of hiccups and the first hiccup happens at one afternoon. Suppose that the donkey hiccups regularly every seconds. At what time does the donkey’s th hiccup occur?
Problem 5
What is the value of
Problem 6
How many of the first ten numbers of the sequence are prime numbers?
Problem 7
For how many values of the constant will the polynomial have two distinct integer roots?
Problem 8
Consider the following sets of elements each: How many of these sets contain exactly two multiples of ?
Problem 9
The sum can be expressed as , where and are positive integers. What is ?
Problem 10
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 11
All the high schools in a large school district are involved in a fundraiser selling T-shirts. Which of the choices below is logically equivalent to the statement "No school bigger than Euclid HS sold more T-shirts than Euclid HS"?
All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS.
No school that sold more T-shirts than Euclid HS is bigger than Euclid HS.
All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS.
All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS.
All schools smaller than Euclid HS sold more T-shirts than Euclid HS.
Problem 12
A pair of fair -sided dice is rolled times. What is the least value of such that the probability that the sum of the numbers face up on a roll equals at least once is greater than ?
Problem 13
The positive difference between a pair of primes is equal to , and the positive difference between the cubes of the two primes is . What is the sum of the digits of the least prime that is greater than those two primes?
Problem 14
Suppose that is a subset of such that the sum of any two (not necessarily distinct) elements of is never an element of What is the maximum number of elements may contain?
Problem 15
Let be the sum of the first terms of an arithmetic sequence that has a common difference of . The quotient does not depend on . What is ?
Problem 16
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 17
One of the following numbers is not divisible by any prime number less than Which is it?
Problem 18
Consider systems of three linear equations with unknowns , , and , where each of the coefficients is either or and the system has a solution other than . For example, one such system is with a nonzero solution of . How many such systems of equations are there? (The equations in a system need not be distinct, and two systems containing the same equations in a different order are considered different.)
Problem 19
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 20
Let be a rhombus with . Let be the midpoint of , and let be the point on such that is perpendicular to . What is the degree measure of ?
Problem 21
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 22
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 23
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 24
Consider functions that satisfy for all real numbers and . Of all such functions that also satisfy the equation , what is the greatest possible value of
Problem 25
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
See also
2022 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by 2022 AMC 10A Problems |
Followed by 2023 AMC 10A Problems | |
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All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.