Difference between revisions of "2021 AMC 10B Problems"
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− | {{AMC10 Problems|year=2021|ab=B}} | + | {{AMC10 2021 Spring Problems|year=2021|ab=B}} |
==Problem 1== | ==Problem 1== | ||
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==Problem 2== | ==Problem 2== | ||
− | What is the value of <math>\sqrt{\left(3-2\sqrt{3}\right)^2}+\sqrt{\left(3+2\sqrt{3}\right)^2}</math>? | + | What is the value of <math>\sqrt{\left(3-2\sqrt{3}\right)^2}+\sqrt{\left(3+2\sqrt{3}\right)^2}</math>? |
<math>\textbf{(A)} ~0 \qquad\textbf{(B)} ~4\sqrt{3}-6 \qquad\textbf{(C)} ~6 \qquad\textbf{(D)} ~4\sqrt{3} \qquad\textbf{(E)} ~4\sqrt{3} + 6</math> | <math>\textbf{(A)} ~0 \qquad\textbf{(B)} ~4\sqrt{3}-6 \qquad\textbf{(C)} ~6 \qquad\textbf{(D)} ~4\sqrt{3} \qquad\textbf{(E)} ~4\sqrt{3} + 6</math> | ||
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==Problem 20== | ==Problem 20== | ||
− | The figure is constructed from <math>11</math> line segments, each of which has length <math>2</math>. The area of pentagon <math>ABCDE</math> can be written | + | The figure is constructed from <math>11</math> line segments, each of which has length <math>2</math>. The area of pentagon <math>ABCDE</math> can be written as <math>\sqrt{m} + \sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers. What is <math>m + n ?</math> |
<asy> | <asy> | ||
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− | <math>\textbf{(A)} ~2 \qquad\textbf{(B)} ~1+\frac{2}{3}\sqrt{3} \qquad\textbf{(C)} ~\ | + | <math>\textbf{(A)} ~2 \qquad\textbf{(B)} ~1+\frac{2}{3}\sqrt{3} \qquad\textbf{(C)} ~\frac{13}{6} \qquad\textbf{(D)} ~1 + \frac{3}{4}\sqrt{3} \qquad\textbf{(E)} ~\frac{7}{3}</math> |
[[2021 AMC 10B Problems/Problem 21|Solution]] | [[2021 AMC 10B Problems/Problem 21|Solution]] | ||
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==See also== | ==See also== | ||
− | {{AMC10 box|year=2021|ab=B|before=[[2021 AMC 10A]]|after=[[ | + | {{AMC10 box|year=2021|ab=B|before=[[2021 AMC 10A]]|after=[[2021 Fall AMC 10A]]}} |
* [[AMC 10]] | * [[AMC 10]] | ||
* [[AMC 10 Problems and Solutions]] | * [[AMC 10 Problems and Solutions]] |
Latest revision as of 12:52, 5 November 2023
2021 AMC 10B (Answer Key) Printable versions: • AoPS Resources • 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
How many integer values of satisfy
?
Problem 2
What is the value of ?
Problem 3
In an after-school program for juniors and seniors, there is a debate team with an equal number of students from each class on the team. Among the students in the program,
of the juniors and
of the seniors are on the debate team. How many juniors are in the program?
Problem 4
At a math contest, students are wearing blue shirts, and another
students are wearing yellow shirts. The
students are assigned into
pairs. In exactly
of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts?
Problem 5
The ages of Jonie's four cousins are distinct single-digit positive integers. Two of the cousins' ages multiplied together give , while the other two multiply to
. What is the sum of the ages of Jonie's four cousins?
Problem 6
Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is , and the afternoon class's mean score is
. The ratio of the number of students in the morning class to the number of students in the afternoon class is
. What is the mean of the scores of all the students?
Problem 7
In a plane, four circles with radii and
are tangent to line
at the same point
but they may be on either side of
. Region
consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region
?
Problem 8
Mr. Zhou places all the integers from to
into a
by
grid. He places
in the middle square (eighth row and eighth column) and places other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest number and the least number that appear in the second row from the top?
Problem 9
The point in the
-plane is first rotated counterclockwise by
around the point
and then reflected about the line
. The image of
after these two transformations is at
. What is
Problem 10
An inverted cone with base radius and height
is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of
. What is the height in centimeters of the water in the cylinder?
Problem 11
Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size and shape, with straight cuts parallel to the sides of the pan. Each cut must be made entirely across the pan. Grandma wants to make the same number of interior pieces as pieces along the perimeter of the pan. What is the greatest possible number of brownies she can produce?
Problem 12
Let . What is the ratio of the sum of the odd divisors of
to the sum of the even divisors of
?
Problem 13
Let be a positive integer and
be a digit such that the value of the numeral
in base
equals
, and the value of the numeral
in base
equals the value of the numeral
in base six. What is
Problem 14
Three equally spaced parallel lines intersect a circle, creating three chords of lengths and
. What is the distance between two adjacent parallel lines?
Problem 15
The real number satisfies the equation
. What is the value of
Problem 16
Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, ,
, and
are all uphill integers, but
,
, and
are not. How many uphill integers are divisible by
?
Problem 17
Ravon, Oscar, Aditi, Tyrone, and Kim play a card game. Each person is given cards out of a set of
cards numbered
The score of a player is the sum of the numbers of their cards. The scores of the players are as follows: Ravon--
Oscar--
Aditi--
Tyrone--
Kim--
Which of the following statements is true?
Problem 18
A fair -sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number?
Problem 19
Suppose that is a finite set of positive integers. If the greatest integer in
is removed from
, then the average value (arithmetic mean) of the integers remaining is
. If the least integer in
is also removed, then the average value of the integers remaining is
. If the greatest integer is then returned to the set, the average value of the integers rises to
. The greatest integer in the original set
is
greater than the least integer in
. What is the average value of all the integers in the set
?
Problem 20
The figure is constructed from line segments, each of which has length
. The area of pentagon
can be written as
, where
and
are positive integers. What is
Problem 21
A square piece of paper has side length and vertices
and
in that order. As shown in the figure, the paper is folded so that vertex
meets edge
at point
, and edge
at point
. Suppose that
. What is the perimeter of triangle
Problem 22
Ang, Ben, and Jasmin each have blocks, colored red, blue, yellow, white, and green; and there are
empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives
blocks all of the same color is
, where
and
are relatively prime positive integers. What is
Problem 23
A square with side length is colored white except for
black isosceles right triangular regions with legs of length
in each corner of the square and a black diamond with side length
in the center of the square, as shown in the diagram. A circular coin with diameter
is dropped onto the square and lands in a random location where the coin is completely contained within the square. The probability that the coin will cover part of the black region of the square can be written as
, where
and
are positive integers. What is
?
Problem 24
Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes and
can be changed into any of the following by one move:
or
Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?
Problem 25
Let be the set of lattice points in the coordinate plane, both of whose coordinates are integers between
and
, inclusive. Exactly
points in
lie on or below a line with equation
. The possible values of
lie in an interval of length
, where
and
are relatively prime positive integers. What is
See also
2021 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by 2021 AMC 10A |
Followed by 2021 Fall AMC 10A | |
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 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.