2021 Fall AMC 10B Problems

2021 Fall AMC 10B (Answer Key) Printable versions: • Fall AoPS Resources • Fall PDF

Instructions This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the SAT if before 2006. No problems on the test will require the use of a calculator). Figures are not necessarily drawn to scale. You will have 75 minutes working time to complete the test.

1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25

Problem 1

What is the value of

Solution

Problem 2

What is the area of the shaded figure shown below?

Solution

Problem 3

The expression is equal to the fraction in which and are positive integers whose greatest common divisor is . What is

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

Solution

Problem 5

Let . Which of the following is equal to

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

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?

Solution

Problem 8

The largest prime factor of is because . What is the sum of the digits of the greatest prime number that is a divisor of ?

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?

Solution

Problem 10

Fourty 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?

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?

Solution

Problem 12

Which of the following conditions is sufficient to guarantee that integers , , and satisfy the equation

and

and

and

and

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?

import olympiad;
pair A,B,C,D,E,F,G,H,I,J,K;
A = origin; B = (0.25,0); C=(1.25,0); D=(1.5,0); E = (0.25,1); F=(0.4166666667,1); G=(1.08333333333,1); H=(1.25,1); I=(0.4166666667,1.66666666667); J=(1.08333333333,1.666666666667); K=(0.75,3);
draw(A--D--K--cycle);
draw(B--C--E--H--cycle);
draw(F--G--I--J--cycle);



<math>(\textbf{A})\: 19\frac14\qquad(\textbf{B}) \: 20\frac14\qquad(\textbf{C}) \: 21 \frac34\qquad(\textbf{D}) \: 22\frac12\qquad(\textbf{E}) \: 23\frac34</math>

[[2021 Fall AMC 10B Problems/Problem 13|Solution]]

==Problem 14==
Una rolls <math>6</math> standard <math>6</math>-sided dice simultaneously and calculates the product of the <math>6{ }</math> numbers obtained. What is the probability that the product is divisible by <math>4?</math>

<math>\textbf{(A)}\: \frac34\qquad\textbf{(B)} \: \frac{57}{64}\qquad\textbf{(C)} \: \frac{59}{64}\qquad\textbf{(D)} \: \frac{187}{192}\qquad\textbf{(E)} \: \frac{63}{64}</math>

[[2021 Fall AMC 10B Problems/Problem 14|Solution]]

==Problem 15==
In square <math>ABCD</math>, points <math>P</math> and <math>Q</math> lie on <math>\overline{AD}</math> and <math>\overline{AB}</math>, respectively. Segments <math>\overline{BP}</math> and <math>\overline{CQ}</math> intersect at right angles at <math>R</math>, with <math>BR=6</math> and <math>PR=7</math>. What is the area of the square?

<asy>
size(170);
defaultpen(linewidth(0.6)+fontsize(10));
real r = 3.5;
pair A = origin, B = (5,0), C = (5,5), D = (0,5), P = (0,r), Q = (5-r,0),
R = intersectionpoint(B--P,C--Q);
draw(A--B--C--D--A^^B--P^^C--Q^^rightanglemark(P,R,C,7));
dot("$A$",A,S);
dot("$B$",B,S);
dot("$C$",C,N);
dot("$D$",D,N);
dot("$Q$",Q,S);
dot("$P$",P,W);
dot("$R$",R,1.3*S);
label("$7$",(P+R)/2,NE);
label("$6$",(R+B)/2,NE);
 (Error making remote request. Unknown error_msg)

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?

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

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 ?

https://d2fr23tawm9iun.cloudfront.net/images/contests/00301-0c73c4b87027afe833e737ee22eee4251f7ac624/files/2021-AMC10B-q18-fig.png

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

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?

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?

Solution

Problem 22

For each integer , let be the sum of all products , where and are integers and . What is the sum of the 10 least values of such that is divisible by ?

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?

Solution

Problem 24

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.)

Solution

Problem 25

A rectangle with side lengths 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

Solution

See also

• AMC 10
• AMC 10 Problems and Solutions
• Mathematics competitions
• Mathematics competition resources

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.