Difference between revisions of "2021 Fall AMC 12B Problems"
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==Problem 15== | ==Problem 15== | ||
− | + | Three identical square sheets of paper each with side length <math>6</math> are stacked on top of each other. The middle sheet is rotated clockwise <math>30^\circ</math> about its center and the top sheet is rotated clockwise <math>60^\circ</math> about its center, resulting in the <math>24</math>-sided polygon shown in the figure below. The area of this polygon can be expressed in the form <math>a-b\sqrt{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers, and <math>c</math> is not divisible by the square of any prime. What is <math>a+b+c</math>? | |
− | + | IMAGE | |
+ | <math>(\textbf{A})\: 75\qquad(\textbf{B}) \: 93\qquad(\textbf{C}) \: 96\qquad(\textbf{D}) \: 129\qquad(\textbf{E}) \: 147</math> | ||
[[2021 Fall AMC 12B Problems/Problem 15|Solution]] | [[2021 Fall AMC 12B Problems/Problem 15|Solution]] | ||
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==Problem 19== | ==Problem 19== | ||
− | + | Regular polygons with <math>5</math>, <math>6</math>, <math>7</math>, and <math>8</math> 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? | |
− | <math>\textbf{ | + | <math>(\textbf{A})\: 52\qquad(\textbf{B}) \: 56\qquad(\textbf{C}) \: 60\qquad(\textbf{D}) \: 64\qquad(\textbf{E}) \: 68</math> |
[[2021 Fall AMC 12B Problems/Problem 19|Solution]] | [[2021 Fall AMC 12B Problems/Problem 19|Solution]] | ||
==Problem 20== | ==Problem 20== | ||
− | + | A cube is constructed from <math>4</math> white unit cubes and <math>4</math> blue unit cubes. How many different ways are there to construct the <math>2 \times 2 \times 2</math> cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.) | |
− | <math>\textbf{ | + | <math>(\textbf{A})\: 7\qquad(\textbf{B}) \: 8\qquad(\textbf{C}) \: 9\qquad(\textbf{D}) \: 10\qquad(\textbf{E}) \: 11</math> |
[[2021 Fall AMC 12B Problems/Problem 20|Solution]] | [[2021 Fall AMC 12B Problems/Problem 20|Solution]] |
Revision as of 20:43, 23 November 2021
2021 Fall AMC 12B (Answer Key) Printable versions: • Fall AoPS Resources • Fall PDF | ||
Instructions
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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
What is the area of the shaded figure shown below?
Problem 3
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
Problem 4
Let . Which of the following is equal to
Problem 5
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?
Problem 6
The largest prime factor of is , because . What is the sum of the digits of the largest prime factor of ?
Problem 7
Which of the following conditions is sufficient to guarantee that integers , , and satisfy the equation
and and and and
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
Problem 10
Problem 11
Una rolls standard -sided dice simultaneously and calculates the product of the numbers obtained. What is the probability that the product is divisible by
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
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 ?
IMAGE
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
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?
Problem 20
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.)
Problem 21
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 Fall AMC 12B Problems |
Followed by [[2021 Fall AMC 12A Problems/Problem {{{num-a}}}|Problem {{{num-a}}}]] |
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.