Difference between revisions of "2024 AMC 10B Problems"
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<math>\textbf{(A) } 9 \qquad\textbf{(B) } 10 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 51 \qquad\textbf{(E) } 100</math> | <math>\textbf{(A) } 9 \qquad\textbf{(B) } 10 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 51 \qquad\textbf{(E) } 100</math> | ||
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+ | [[2024 AMC 10B Problems/Problem 12|Solution]] | ||
==Problem 13== | ==Problem 13== | ||
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==Problem 14== | ==Problem 14== | ||
A dartboard is the region B in the coordinate plane consisting of points (x,y) such | A dartboard is the region B in the coordinate plane consisting of points (x,y) such | ||
− | that |x|+|y|≤8 . A target T is the region where(x^2+y^2-25)^2 ≤ 49 . A dart is | + | that |x|+|y|≤8 . A target T is the region where (x^2+y^2-25)^2 ≤ 49 . A dart is |
− | thrown at a random point in B. The probability that the dart lands in T can be | + | thrown at a random point in <math>B</math>. The probability that the dart lands in T can be |
expressed as (m/n)π, where and are relatively prime positive integers. What | expressed as (m/n)π, where and are relatively prime positive integers. What | ||
is m+n ? | is m+n ? | ||
+ | <math>\textbf{(A) } 39 \qquad\textbf{(B) } 71 \qquad\textbf{(C) } 73 \qquad\textbf{(D) } 75 \qquad\textbf{(E) } 135</math> | ||
− | + | [[2024 AMC 10B Problems/Problem 14|Solution]] | |
==Problem 15== | ==Problem 15== | ||
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<math>\textbf{(A) } 180 \qquad\textbf{(B) } 361 \qquad\textbf{(C) } 420 \qquad\textbf{(D) } 431 \qquad\textbf{(E) } 720</math> | <math>\textbf{(A) } 180 \qquad\textbf{(B) } 361 \qquad\textbf{(C) } 420 \qquad\textbf{(D) } 431 \qquad\textbf{(E) } 720</math> | ||
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+ | [[2024 AMC 10B Problems/Problem 17|Solution]] | ||
==Problem 18== | ==Problem 18== | ||
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<math>\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } 25 \qquad\textbf{(E) } 125</math> | <math>\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } 25 \qquad\textbf{(E) } 125</math> | ||
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+ | [[2024 AMC 10B Problems/Problem 18|Solution]] | ||
==Problem 19== | ==Problem 19== | ||
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<math>\textbf{(A) } 60 \qquad\textbf{(B) } 72 \qquad\textbf{(C) } 90 \qquad\textbf{(D) } 108 \qquad\textbf{(E) } 120</math> | <math>\textbf{(A) } 60 \qquad\textbf{(B) } 72 \qquad\textbf{(C) } 90 \qquad\textbf{(D) } 108 \qquad\textbf{(E) } 120</math> | ||
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+ | [[2024 AMC 10B Problems/Problem 20|Solution]] | ||
==Problem 21== | ==Problem 21== | ||
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<math>\textbf{(A) } 5 \qquad\textbf{(B) } 6 \qquad\textbf{(C) } 7 \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 9</math> | <math>\textbf{(A) } 5 \qquad\textbf{(B) } 6 \qquad\textbf{(C) } 7 \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 9</math> | ||
+ | |||
+ | [[2024 AMC 10B Problems/Problem 22|Solution]] | ||
==Problem 23== | ==Problem 23== | ||
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==Problem 24== | ==Problem 24== | ||
+ | Let | ||
+ | <cmath>P(m)=\frac{m}{2}+\frac{m^2}{4}+\frac{m^4}{8}+\frac{m^8}{8}</cmath> | ||
+ | How many of the values <math>P(2022)</math>, <math>P(2023)</math>, <math>P(2024)</math>, and <math>P(2025)</math> are integers? | ||
+ | |||
+ | <math>\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4</math> | ||
+ | |||
+ | [[2024 AMC 10B Problems/Problem 24|Solution]] | ||
==Problem 25== | ==Problem 25== |
Latest revision as of 05:39, 14 November 2024
2024 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 |
Problems to be added after the competition.
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
In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the right. How many people are in line?
Problem 2
What is
Problem 3
For how many integer values of is
Problem 4
Balls numbered 1, 2, 3, ... are deposited in 5 bins, labeled A, B, C, D, and E, using the following procedure. Ball 1 is deposited in bin A, and balls 2 and 3 are deposited in bin B. The next 3 balls are deposited in bin C, the next 4 in bin D, and so on, cycling back to bin A after balls are deposited in bin E. (For example, balls numbered 22, 23, ..., 28 are deposited in bin B at step 7 of this process.) In which bin is ball 2024 deposited?
Problem 5
In the following expression, Melanie changed some of the plus signs to minus signs: When the new expression was evaluated, it was negative. What is the least number of plus signs that Melanie could have changed to minus signs?
Problem 6
A rectangle has integer length sides and an area of 2024. What is the least possible perimeter of the rectangle?
Problem 7
What is the remainder when is divided by ?
Problem 8
Let be the product of all the positive integer divisors of . What is the units digit of ?
Problem 9
Real numbers and have arithmetic mean 0. The arithmetic mean of and is 10. What is the arithmetic mean of and ?
Problem 10
Quadrilateral is a parallelogram, and is the midpoint of the side . Let be the intersection of lines and . What is the ratio of the area of quadrilateral to the area of triangle ?
Problem 11
Problem 12
A group of students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and, for every pair of students and , student speaks some language that student does not speak, and student speaks some language that student does not speak. What is the least possible total number of languages spoken by all the students?
Problem 13
Positive integers and satisfy the equation . What is the minimum possible value of .
Problem 14
A dartboard is the region B in the coordinate plane consisting of points (x,y) such that |x|+|y|≤8 . A target T is the region where (x^2+y^2-25)^2 ≤ 49 . A dart is thrown at a random point in . The probability that the dart lands in T can be expressed as (m/n)π, where and are relatively prime positive integers. What is m+n ?
Problem 15
Problem 16
Problem 17
In a race among 5 snails, there is at most one tie, but that tie can involve any number of snails. For example, the result of the race might be that Dazzler is first; Abby, Cyrus, and Elroy are tied for second, and Bruna is fifth. How many different results of the race are possible?
Problem 18
How many different remainders can result when the th power of an integer is divided by ?
Problem 19
Problem 20
Three different pairs of shoes are placed in a row so that no left shoe is next to a right shoe from a different pair. In how many ways can these six shoes be lined up?
Problem 21
Problem 22
A group of people will be partitioned into indistinguishable -person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as , where and are positive integers and is not divisible by . What is ?
Problem 23
The Fibonacci numbers are defined by and for What is
Problem 24
Let How many of the values , , , and are integers?
Problem 25
See also
2024 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by 2024 AMC 10A Problems |
Followed by 2025 AMC 10A 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 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.