2021 Fall AMC 12A Problems
2021 Fall AMC 12A (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
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 4
The six-digit number is prime for only one digit What is
Problem 5
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 6
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 7
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 8
Let be the least common multiple of all the integers through inclusive. Let be the least common multiple of and What is the value of
Problem 9
A right rectangular prism whose surface area and volume are numerically equal has edge lengths and What is
Problem 10
The base-nine representation of the number is What is the remainder when is divided by
Problem 11
Consider two concentric circles of radius and The larger circle has a chord, half of which lies inside the smaller circle. What is the length of the chord in the larger circle?
Problem 12
What is the number of terms with rational coefficients among the terms in the expansion of
Problem 13
The angle bisector of the acute angle formed at the origin by the graphs of the lines and has equation What is
Problem 14
In the figure, equilateral hexagon has three nonadjacent acute interior angles that each measure . The enclosed area of the hexagon is . What is the perimeter of the hexagon?
Problem 15
Recall that the conjugate of the complex number , where and are real numbers and , is the complex number . For any complex number , let . The polynomial has four complex roots: , , , and . Let be the polynomial whose roots are , , , and , where the coefficients and are complex numbers. What is
Problem 16
An organization has employees, of whom have a brand A computer while the other have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other if their computers are directly connected by a cable or by relaying messages through a series of connected computers. Initially, no computer is connected to any other. A technician arbitrarily selects one computer of each brand and installs a cable between them, provided there is not already a cable between that pair. The technician stops once every employee can communicate with each other. What is the maximum possible number of cables used?
Problem 17
For how many ordered pairs of positive integers does neither nor have two distinct real solutions?
Problem 18
Each of balls is tossed independently and at random into one of 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 19
Let be the least real number greater than such that , where the arguments are in degrees. What is rounded up to the closest integer?
Problem 20
For each positive integer , let be twice the number of positive integer divisors of , and for , let . For how many values of is
Problem 21
Let be an isosceles trapezoid with and . Points and lie on diagonal with between and , as shown in the figure. Suppose , , , and . What is the area of ?
Problem 22
Azar and Carl play a game of tic-tac-toe. Azar places an in one of the boxes in a -by- array of boxes, then Carl places an in one of the remaining boxes. After that, Azar places an in one of the remaining boxes, and so on until all boxes are filled or one of the players has of their symbols in a row—horizontal, vertical, or diagonal—whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third . How many ways can the board look after the game is over?
Problem 23
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 ?
Problem 24
Convex quadrilateral has and In some order, the lengths of the four sides form an arithmetic progression, and side is a side of maximum length. The length of another side is What is the sum of all possible values of ?
Problem 25
Let be an odd integer, and let denote the number of quadruples of distinct integers with for all such that divides . There is a polynomial such that for all odd integers . What is
See Also
2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
Preceded by 2021 AMC 12B Problems |
Followed by 2021 Fall AMC 12B Problems |
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All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.