Difference between revisions of "2021 Fall AMC 12A Problems"
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==Problem 25== | ==Problem 25== | ||
− | Let <math>m\ge 5</math> be an odd integer, and let <math>D(m)</math> denote the number of quadruples <math> | + | Let <math>m\ge 5</math> be an odd integer, and let <math>D(m)</math> denote the number of quadruples <math>(a_1, a_2, a_3, a_4)</math> of distinct integers with <math>1\le a_i \le m</math> for all <math>i</math> such that <math>m</math> divides <math>a_1+a_2+a_3+a_4</math>. There is a polynomial |
<cmath>q(x) = c_3x^3+c_2x^2+c_1x+c_0</cmath>such that <math>D(m) = q(m)</math> for all odd integers <math>m\ge 5</math>. What is <math>c_1?</math> | <cmath>q(x) = c_3x^3+c_2x^2+c_1x+c_0</cmath>such that <math>D(m) = q(m)</math> for all odd integers <math>m\ge 5</math>. What is <math>c_1?</math> | ||
Revision as of 01:39, 24 November 2021
2021 Fall AMC 12A (Answer Key) Printable versions: • Fall AoPS Resources • Fall PDF | ||
Instructions
| ||
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 |
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 sin
= sin
, 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 3-by-3 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 |
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 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.