2021 Fall AMC 12B Problems
2021 Fall AMC 12B (Answer Key) Printable versions: • Fall AoPS Resources • Fall 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
What is the value of
Problem 2
What is the area of the shaded figure shown below?
Problem 3
At noon on a certain day, Minneapolis is degrees warmer than St. Louis. At the temperature in Minneapolis has fallen by degrees while the temperature in St. Louis has risen by degrees, at which time the temperatures in the two cities differ by degrees. What is the product of all possible values of
Problem 4
Let . Which of the following is equal to
Problem 5
Call a fraction , not necessarily in the simplest form, special if and are positive integers whose sum is . How many distinct integers can be written as the sum of two, not necessarily different, special fractions?
Problem 6
The largest prime factor of is because . What is the sum of the digits of the greatest prime number that is a divisor of ?
Problem 7
Which of the following conditions is sufficient to guarantee that integers , , and satisfy the equation
and
and
and
and
Problem 8
The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle's height to the base. What is the measure, in degrees, of the vertex angle of this triangle?
Problem 9
Triangle is equilateral with side length . Suppose that is the center of the inscribed circle of this triangle. What is the area of the circle passing through , , and ?
Problem 10
What is the sum of all possible values of between and such that the triangle in the coordinate plane whose vertices are is isosceles?
Problem 11
Una rolls standard -sided dice simultaneously and calculates the product of the numbers obtained. What is the probability that the product is divisible by
Problem 12
For a positive integer, let be the quotient obtained when the sum of all positive divisors of is divided by For example, What is
Problem 13
Let What is the value of
Problem 14
Suppose that , and are polynomials with real coefficients, having degrees , , and , respectively, and constant terms , , and , respectively. Let be the number of distinct complex numbers that satisfy the equation . What is the minimum possible value of ?
Problem 15
Three identical square sheets of paper each with side length are stacked on top of each other. The middle sheet is rotated clockwise about its center and the top sheet is rotated clockwise about its center, resulting in the -sided polygon shown in the figure below. The area of this polygon can be expressed in the form , where , , and are positive integers, and is not divisible by the square of any prime. What is ?
IMAGE
Problem 16
Suppose , , are positive integers such that and What is the sum of all possible distinct values of ?
Problem 17
A bug starts at a vertex of a grid made of equilateral triangles of side length . At each step the bug moves in one of the possible directions along the grid lines randomly and independently with equal probability. What is the probability that after moves the bug never will have been more than unit away from the starting position?
Problem 18
Set , and for let be determined by the recurrence
This sequence tends to a limit; call it . What is the least value of such that
Problem 19
Regular polygons with , , , and sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect?
Problem 20
A cube is constructed from white unit cubes and blue unit cubes. How many different ways are there to construct the cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)
Problem 21
For real numbers , let where . For how many values of with does
Problem 22
Right triangle has side lengths , , and .
A circle centered at is tangent to line at and passes through . A circle centered at is tangent to line at and passes through . What is ?
Problem 23
What is the average number of pairs of consecutive integers in a randomly selected subset of distinct integers chosen from the set ? (For example the set has pairs of consecutive integers.)
Problem 24
Triangle has side lengths , and . The bisector of intersects in point , and intersects the circumcircle of in point . The circumcircle of intersects the line in points and . What is ?
Problem 25
For a positive integer, let be the sum of the remainders when is divided by , , , , , , , , and . For example, . How many two-digit positive integers satisfy
See also
2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
Preceded by 2021 Fall AMC 12A Problems |
Followed by 2022 AMC 12A Problems |
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The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.