ΒΆ 2021 AMC 10B_Fall Problems and Solutions
Problem Set Workbook
Access the downloadable workbook for 2021 AMC 10B_Fall problems here.
Discussion Forum
Engage in discussion about the 2021 AMC 10B_Fall math contest by visiting Random Math AMC 10B_Fall 2021 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2021 AMC 10B_Fall problems, please refer below:
Problem 1: What is the value of
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 2: What is the area of the shaded figure shown below?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 3: The expression is equal to the fraction , where and are positive integers whose greatest common divisor is . What is ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 4: 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 ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 5: Let . Which of the following is equal to ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 6: The least positive integer with exactly distinct positive divisors can be written in the form , where and are integers and is not a divisor of . What is ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 7: 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?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 8: The largest prime factor of is , because . What is the sum of the digits of the largest prime factor of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 9: The knights in a certain kingdom come in two colors. of them are red, and the rest are blue. Furthermore, of the knights are magical, and the fraction of red knights who are magical is times the fraction of blue knights who are magical. What fraction of red knights are magical?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 10: Forty slips of paper numbered to are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, "I can't tell who has the larger number." Then Bob says, "I know who has the larger number." Alice says, "You do? Is your number prime?" Bob replies, "Yes." Alice says, "In that case, if I multiply your number by and add my number, the result is a perfect square. " What is the sum of the two numbers drawn from the hat?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 11: A regular hexagon of side length is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. What is the area of the region bounded by these reflected arcs?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 12: Which of the following conditions is sufficient to guarantee that integers , and satisfy the equation
Answer Choices:
A. and
B. and
C. and
D. and
E.
Solution:
Problem 13: A square with side length is inscribed in an isosceles triangle with one side of the square along the base of the triangle. A square with side length has two vertices on the other square and the other two on sides of the triangle, as shown. What is the area of the triangle?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 14: Una rolls standard -sided dice simultaneously and calculates the product of the numbers obtained. What is the probability that the product is divisible by ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 15: In square , points and lie on and , respectively. Segments and intersect at right angles at , with and . What is the area of the square?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 16: Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. What is the expected number of balls that occupy their original positions after these two successive transpositions?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 17: Distinct lines and lie in the -plane. They intersect at the origin. Point is reflected about line to point , and then is reflected about line to point . The equation of line is , and the coordinates of are . What is the equation of line ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 18: 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 ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 19: Let be the positive integer , a -digit number where each digit is a . Let be the leading digit of the th root of . What is
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 20: In a particular game, each of players rolls a standard -sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again and this process will continue until one player wins. Hugo is one of the players in this game. What is the probability that Hugo's first roll was a , given that he won the game?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 21: 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?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 22: For each integer , let be the sum of all products , where and are integers and . What is the sum of the least values of such that is divisible by ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 23: Each of the sides and the diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides have the same color?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 24: A cube is constructed from white unit cubes and black 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.)
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 25: A rectangle with side length and , a square with side length , and a rectangle are inscribed inside a larger square as shown. The sum of all possible values for the area of can be written in the form , where and are relatively prime positive integers. 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