2021 Fall AMC 10A Problems
2021 Fall AMC 10A (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
Menkara has a index card. If she shortens the length of one side of this card by inch, the card would have area square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by inch?
Problem 3
What is the maximum number of balls of clay of radius that can completely fit inside a cube of side length assuming the balls can be reshaped but not compressed before they are packed in the cube?
Problem 4
Mr. Lopez has a choice of two routes to get to work. Route A is miles long, and his average speed along this route is miles per hour. Route B is miles long, and his average speed along this route is miles per hour, except for a -mile stretch in a school zone where his average speed is miles per hour. By how many minutes is Route B quicker than Route A?
Problem 5
The six-digit number is prime for only one digit What is
Problem 6
Elmer the emu takes equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in equal leaps. The telephone poles are evenly spaced, and the st pole along this road is exactly one mile ( feet) from the first pole. How much longer, in feet, is Oscar's leap than Elmer's stride?
Problem 7
As shown in the figure below, point lies on the opposite half-plane determined by line from point so that . Point lies on so that , and is a square. What is the degree measure of ?
Problem 8
A two-digit positive integer is said to be if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly?
Problem 9
When a certain unfair die is rolled, an even number is times as likely to appear as an odd number. The die is rolled twice. What is the probability that the sum of the numbers rolled is even?
Problem 10
A school has students and teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are and . Let be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is ?
Problem 11
Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate faster than the ship. She counts equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts steps of the same size from the front of the ship to the back. In terms of Emily's equal steps, what is the length of the ship?
Problem 12
The base-nine representation of the number is What is the remainder when is divided by
Problem 13
Each of balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other balls?
Problem 14
How many ordered pairs of real numbers satisfy the following system of equations?
Problem 15
Isosceles triangle has , and a circle with radius is tangent to line at and to line at . What is the area of the circle that passes through vertices , , and
Problem 16
The graph of is symmetric about which of the following? (Here is the greatest integer not exceeding .)
Problem 17
An architect is building a structure that will place vertical pillars at the vertices of regular hexagon , which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at , , and are , , and meters, respectively. What is the height, in meters, of the pillar at ?
Problem 18
A farmer's rectangular field is partitioned into by grid of rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field?
Problem 19
A disk of radius rolls all the way around the inside of a square of side length and sweeps out a region of area . A second disk of radius rolls all the way around the outside of the same square and sweeps out a region of area . The value of can be written as , where , and are positive integers and and are relatively prime. What is ?
Problem 20
How many ordered pairs of positive integers exist where both and do not have distinct, real solutions?
Problem 21
Each of the balls is tossed independently and at random into one of the bins. Let be the probability that some bin ends up with balls, another with balls, and the other three with balls each. Let be the probability that every bin ends up with balls. What is ?
Problem 22
Inside a right circular cone with base radius and height are three congruent spheres with radius . Each sphere is tangent to the other two spheres and also tangent to the base and side of the cone. What is ?
Problem 23
For each positive integer , let be twice the number of positive integer divisors of , and for , let . For how many values of is
Problem 24
Each of the edges of a cube is labeled or . Two labelings are considered different even if one can be obtained from the other by a sequence of one or more rotations and/or reflections. For how many such labelings is the sum of the labels on the edges of each of the faces of the cube equal to ?
Problem 25
A quadratic polynomial with real coefficients and leading coefficient is called if the equation is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial for which the sum of the roots is maximized. What is ?
See Also
2021 Fall AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by 2021 AMC 10B |
Followed by 2021 Fall AMC 10B | |
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 | ||
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The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.