Difference between revisions of "2020 AMC 12B Problems"
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==Problem 21== | ==Problem 21== | ||
− | How many positive integers <math>n</math> satisfy<cmath>\ | + | How many positive integers <math>n</math> satisfy<cmath>\frac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?</cmath>(Recall that <math>\lfloor x\rfloor</math> is the greatest integer not exceeding <math>x</math>.) |
<math>\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32</math> | <math>\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32</math> |
Revision as of 09:56, 11 April 2020
2020 AMC 12B (Answer Key) Printable version: | AoPS Resources • 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 in simplest form of the following expression?
Problem 2
What is the value of the following expression?
Problem 3
The ratio of to is , the ratio of to is , and the ratio of to is . What is the ratio of to ?
Problem 4
The acute angles of a right triangle are and , where and both and are prime numbers. What is the least possible value of ?
Problem 5
Teams and are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team has won of its games and team has won of its games. Also, team has won more games and lost more games than team How many games has team played?
Problem 6
For all integers the value of is always which of the following?
Problem 7
Two nonhorizontal, non vertical lines in the -coordinate plane intersect to form a angle. One line has slope equal to times the slope of the other line. What is the greatest possible value of the product of the slopes of the two lines?
Problem 8
How many ordered pairs of integers satisfy the equation
Problem 9
A three-quarter sector of a circle of radius inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?
Problem 10
In unit square the inscribed circle intersects at and intersects at a point different from What is
Problem 11
As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region—inside the hexagon but outside all of the semicircles?
Problem 12
Let be a diameter in a circle of radius Let be a chord in the circle that intersects at a point such that and What is
Problem 13
Which of the following is the value of
Problem 14
Bela and Jenn play the following game on the closed interval of the real number line, where is a fixed integer greater than . They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval . Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?
Problem 15
There are 10 people standing equally spaced around a circle. Each person knows exactly 3 of the other 9 people: the 2 people standing next to her or him, as well as the person directly across the circle. How many ways are there for the 10 people to split up into 5 pairs so that the members of each pair know each other?
Problem 16
An urn contains one red ball and one blue ball. A box of extra red and blue balls lie nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?
Problem 17
How many polynomials of the form , where , , , and are real numbers, have the property that whenever is a root, so is ? (Note that )
Problem 18
In square , points and lie on and , respectively, so that Points and lie on and , respectively, and points and lie on so that and . See the figure below. Triangle , quadrilateral , quadrilateral , and pentagon each has area What is ?
Problem 19
Square in the coordinate plane has vertices at the points and Consider the following four transformations: a rotation of counterclockwise around the origin; a rotation of clockwise around the origin; a reflection across the -axis; and a reflection across the -axis.
Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying and then would send the vertex at to and would send the vertex at to itself. How many sequences of transformations chosen from will send all of the labeled vertices back to their original positions? (For example, is one sequence of transformations that will send the vertices back to their original positions.)
Problem 20
Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in appearance?
Problem 21
How many positive integers satisfy(Recall that is the greatest integer not exceeding .)
Problem 22
What is the maximum value of for real values of
Problem 23
How many integers are there such that whenever are complex numbers such that
then the numbers are equally spaced on the unit circle in the complex plane?
Problem 24
Let denote the number of ways of writing the positive integer as a productwhere , the are integers strictly greater than , and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number can be written as , , and , so . What is ?
Problem 25
For each real number with , let numbers and be chosen independently at random from the intervals and , respectively, and let be the probability that
What is the maximum value of
See also
2020 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by 2020 AMC 12A Problems |
Followed by 2021 AMC 12A 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.