Difference between revisions of "2021 AMC 10B Problems"
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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 is <math>\sqrt{m} + \sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers. What is <math>m + n ?</math> | |
+ | [asy] | ||
+ | /* Made by samrocksnature */ | ||
+ | pair A=(-2.4638,4.10658); | ||
+ | pair B=(-4,2.6567453480756127); | ||
+ | pair C=(-3.47132,0.6335248637894945); | ||
+ | pair D=(-1.464483379039766,0.6335248637894945); | ||
+ | pair E=(-0.956630463955801,2.6567453480756127); | ||
+ | pair F=(-2,2); | ||
+ | pair G=(-3,2); | ||
+ | draw(A--B--C--D--E--A); | ||
+ | draw(A--F--A--G); | ||
+ | draw(B--F--C); | ||
+ | draw(E--G--D); | ||
+ | label("A",A,N); | ||
+ | label("B",B,W); | ||
+ | label("C",C,W); | ||
+ | label("D",D,E); | ||
+ | label("E",E,dir(0)); | ||
+ | dot(A^^B^^C^^D^^E^^F^^G); | ||
+ | [/asy] | ||
− | <math>\textbf{(A)} ~ | + | <math>\textbf{(A)} ~20 \qquad\textbf{(B)} ~21 \qquad\textbf{(C)} ~22 \qquad\textbf{(D)} ~23 \qquad\textbf{(E)} ~24</math> |
==Problem 21== | ==Problem 21== |
Revision as of 19:21, 11 February 2021
2021 AMC 10B (Answer Key) Printable versions: • 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
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
Problem 11
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
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
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 [asy] /* Made by samrocksnature */ pair A=(-2.4638,4.10658); pair B=(-4,2.6567453480756127); pair C=(-3.47132,0.6335248637894945); pair D=(-1.464483379039766,0.6335248637894945); pair E=(-0.956630463955801,2.6567453480756127); pair F=(-2,2); pair G=(-3,2); draw(A--B--C--D--E--A); draw(A--F--A--G); draw(B--F--C); draw(E--G--D); label("A",A,N); label("B",B,W); label("C",C,W); label("D",D,E); label("E",E,dir(0)); dot(A^^B^^C^^D^^E^^F^^G); [/asy]
Problem 21
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
Problem 24
Problem 25
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.