Difference between revisions of "2021 AMC 10B Problems"
(2021 AMC 10B Problem 18 wasn't really 2021 AMC 12B Problem 16 (the first one was a combinatorics one, the second one was algebra)) (Tag: Undo) |
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==Problem 2== | ==Problem 2== | ||
− | What is the value of <math>\sqrt{(3-2\sqrt{3})^2}+\sqrt{(3+2\sqrt{3})^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 7== | ==Problem 7== | ||
− | In a plane, four circles with radii <math>1,3,5,</math> and <math>7</math> are tangent to line <math> | + | In a plane, four circles with radii <math>1,3,5,</math> and <math>7</math> are tangent to line <math>\ell</math> at the same point <math>A,</math> but they may be on either side of <math>\ell</math>. Region <math>S</math> consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region <math>S</math>? |
<math>\textbf{(A) }24\pi \qquad \textbf{(B) }32\pi \qquad \textbf{(C) }64\pi \qquad \textbf{(D) }65\pi \qquad \textbf{(E) }84\pi</math> | <math>\textbf{(A) }24\pi \qquad \textbf{(B) }32\pi \qquad \textbf{(C) }64\pi \qquad \textbf{(D) }65\pi \qquad \textbf{(E) }84\pi</math> | ||
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==Problem 17== | ==Problem 17== | ||
− | Ravon, Oscar, Aditi, Tyrone, and Kim play a card game. Each person is given <math>2</math> cards out of a set of <math>10</math> cards numbered <math>1, 2, 3,\ | + | Ravon, Oscar, Aditi, Tyrone, and Kim play a card game. Each person is given <math>2</math> cards out of a set of <math>10</math> cards numbered <math>1,2,3, \dots,10.</math> The score of a player is the sum of the numbers of their cards. The scores of the players are as follows: Ravon--<math>11,</math> Oscar--<math>4,</math> Aditi--<math>7,</math> Tyrone--<math>16,</math> Kim--<math>17.</math> Which of the following statements is true? |
− | <math>\textbf{(A)} | + | <math>\textbf{(A) }\text{Ravon was given card 3.}</math> |
− | <math>\textbf{(B)} | + | |
− | <math>\textbf{(C)} | + | <math>\textbf{(B) }\text{Aditi was given card 3.}</math> |
− | <math>\textbf{(D)} | + | |
− | <math>\textbf{(E)} | + | <math>\textbf{(C) }\text{Ravon was given card 4.}</math> |
+ | |||
+ | <math>\textbf{(D) }\text{Aditi was given card 4.}</math> | ||
+ | |||
+ | <math>\textbf{(E) }\text{Tyrone was given card 7.}</math> | ||
[[2021 AMC 10B Problems/Problem 17|Solution]] | [[2021 AMC 10B Problems/Problem 17|Solution]] | ||
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==Problem 19== | ==Problem 19== | ||
− | Suppose that <math>S</math> is a finite set of positive integers. If the greatest integer in <math>S</math> is removed from <math>S</math>, then the average value (arithmetic mean) of the integers remaining is <math>32</math>. If the least integer | + | Suppose that <math>S</math> is a finite set of positive integers. If the greatest integer in <math>S</math> is removed from <math>S</math>, then the average value (arithmetic mean) of the integers remaining is <math>32</math>. If the least integer in <math>S</math> is also removed, then the average value of the integers remaining is <math>35</math>. If the greatest integer is then returned to the set, the average value of the integers rises to <math>40</math>. The greatest integer in the original set <math>S</math> is <math>72</math> greater than the least integer in <math>S</math>. What is the average value of all the integers in the set <math>S</math>? |
− | <math>\textbf{(A)} | + | <math>\textbf{(A) }36.2 \qquad \textbf{(B) }36.4 \qquad \textbf{(C) }36.6\qquad \textbf{(D) }36.8 \qquad \textbf{(E) }37</math> |
[[2021 AMC 10B Problems/Problem 19|Solution]] | [[2021 AMC 10B Problems/Problem 19|Solution]] | ||
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label("B",B,W); | label("B",B,W); | ||
label("C",C,W); | label("C",C,W); | ||
− | label("D",D, | + | label("D",D,dir(0)); |
label("E",E,dir(0)); | label("E",E,dir(0)); | ||
dot(A^^B^^C^^D^^E^^F^^G); | dot(A^^B^^C^^D^^E^^F^^G); | ||
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label("E",E,NW); | label("E",E,NW); | ||
label("C'",CC,N); | label("C'",CC,N); | ||
+ | dot(C); | ||
+ | dot(E); | ||
</asy> | </asy> | ||
− | <math>\textbf{(A)} ~2 \qquad\textbf{(B)} ~1+\frac{2}{3}\sqrt{3} \qquad\textbf{(C)} ~\sqrt{ | + | <math>\textbf{(A)} ~2 \qquad\textbf{(B)} ~1+\frac{2}{3}\sqrt{3} \qquad\textbf{(C)} ~\sqrt{136} \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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==Problem 23== | ==Problem 23== | ||
− | A square with side length <math>8</math> is colored white except for <math>4</math> black isosceles right triangular regions with legs of length <math>2</math> in each corner of the square and a black diamond with side length <math>2\sqrt{2}</math> in the center of the square, as shown in the diagram. A circular coin with diameter <math>1</math> is dropped onto the square and lands in a random location where the coin is completely contained within the square | + | A square with side length <math>8</math> is colored white except for <math>4</math> black isosceles right triangular regions with legs of length <math>2</math> in each corner of the square and a black diamond with side length <math>2\sqrt{2}</math> in the center of the square, as shown in the diagram. A circular coin with diameter <math>1</math> 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 <math>\frac{1}{196}\left(a+b\sqrt{2}+\pi\right)</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math>? |
− | |||
<asy> | <asy> | ||
/* Made by samrocksnature */ | /* Made by samrocksnature */ | ||
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fill((0,6)--(2,8)--(0,8)--cycle, black); | fill((0,6)--(2,8)--(0,8)--cycle, black); | ||
fill((4,6)--(2,4)--(4,2)--(6,4)--cycle, black); | fill((4,6)--(2,4)--(4,2)--(6,4)--cycle, black); | ||
− | filldraw(circle((2.6,3.31),0. | + | filldraw(circle((2.6,3.31),0.5),gray); |
− | |||
</asy> | </asy> | ||
− | <math>\textbf{(A) }64 \qquad \textbf{(B) }66 \qquad \textbf{(C) }68 \qquad \textbf{(D) }70 \qquad \textbf{(E) }72</math> | + | <math>\textbf{(A)} ~64 \qquad\textbf{(B)} ~66 \qquad\textbf{(C)} ~68 \qquad\textbf{(D)} ~70 \qquad\textbf{(E)} ~72</math> |
[[2021 AMC 10B Problems/Problem 23|Solution]] | [[2021 AMC 10B Problems/Problem 23|Solution]] | ||
==Problem 24== | ==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 <math>4</math> and <math>2</math> can be changed into any of the following by one move: <math>(3,2),(2,1,2),(4),(4,1),(2,2),</math> or <math>(1,1,2).</math> | ||
− | + | <asy> unitsize(4mm); real[] boxes = {0,1,2,3,5,6,13,14,15,17,18,21,22,24,26,27,30,31,32,33}; for(real i:boxes){ draw(box((i,0),(i+1,3))); } draw((8,1.5)--(12,1.5),Arrow()); defaultpen(fontsize(20pt)); label(",",(20,0)); label(",",(29,0)); label(",...",(35.5,0)); </asy> | |
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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? | 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? | ||
− | <math>\textbf{(A)} | + | <math>\textbf{(A) }(6,1,1) \qquad \textbf{(B) }(6,2,1) \qquad \textbf{(C) }(6,2,2)\qquad \textbf{(D) }(6,3,1) \qquad \textbf{(E) }(6,3,2)</math> |
[[2021 AMC 10B Problems/Problem 24|Solution]] | [[2021 AMC 10B Problems/Problem 24|Solution]] |
Latest revision as of 20:21, 4 September 2021
2021 AMC 10B (Answer Key) Printable version: | 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 is , 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 2022 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.