Difference between revisions of "2024 AMC 12B Problems"
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==Problem 2== | ==Problem 2== | ||
+ | What is <math>10! - 7! \cdot 6!</math>? | ||
+ | |||
+ | <math>\textbf{(A) }-120 \qquad\textbf{(B) }0 \qquad\textbf{(C) }120 \qquad\textbf{(D) }600 \qquad\textbf{(E) }720 \qquad</math> | ||
+ | |||
[[2024 AMC 12B Problems/Problem 2|Solution]] | [[2024 AMC 12B Problems/Problem 2|Solution]] | ||
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==Problem 4== | ==Problem 4== | ||
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+ | Balls numbered <math>1,2,3,\ldots</math> are deposited in <math>5</math> bins, labeled <math>A,B,C,D,</math> and <math>E</math>, using the following procedure. Ball <math>1</math> is deposited in bin <math>A</math>, and balls <math>2</math> and <math>3</math> are deposited in <math>B</math>. The next three balls are deposited in bin <math>C</math>, the next <math>4</math> in bin <math>D</math>, and so on, cycling back to bin <math>A</math> after balls are deposited in bin <math>E</math>. (For example, <math>22,23,\ldots,28</math> are deposited in bin <math>B</math> at step 7 of this process.) In which bin is ball <math>2024</math> deposited? | ||
+ | |||
+ | <math>\textbf{(A) }A\qquad\textbf{(B) }B\qquad\textbf{(C) }C\qquad\textbf{(D) }D\qquad\textbf{(E) }E</math> | ||
+ | |||
[[2024 AMC 12B Problems/Problem 4|Solution]] | [[2024 AMC 12B Problems/Problem 4|Solution]] | ||
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==Problem 10== | ==Problem 10== | ||
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+ | A list of 9 real numbers consists of <math>1</math>, <math>2.2 </math>, <math>3.2 </math>, <math>5.2 </math>, <math>6.2 </math>, <math>7</math>, as well as <math>x, y,z</math> with <math>x\leq y\leq z</math>. The range of the list is <math>7</math>, and the mean and median are both positive integers. How many ordered triples <math>(x,y,z)</math> are possible? | ||
+ | |||
+ | <math>\textbf{(A) }1 \qquad\textbf{(B) }2 \qquad\textbf{(C) }3 \qquad\textbf{(D) }4 \qquad\textbf{(E) \text{infinitely many}}\qquad</math> | ||
+ | |||
[[2024 AMC 12B Problems/Problem 10|Solution]] | [[2024 AMC 12B Problems/Problem 10|Solution]] | ||
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==Problem 14== | ==Problem 14== | ||
+ | How many different remainders can result when the <math>100</math>th power of an integer is divided by <math>125</math>? | ||
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+ | <math>\textbf{(A) }1 \qquad\textbf{(B) }2 \qquad\textbf{(C) }5 \qquad\textbf{(D) }25 \qquad\textbf{(E) }125 \qquad</math> | ||
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[[2024 AMC 12B Problems/Problem 14|Solution]] | [[2024 AMC 12B Problems/Problem 14|Solution]] | ||
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==Problem 25== | ==Problem 25== | ||
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+ | Pablo will decorate each of <math>6</math> identical white balls with either a striped or a dotted pattern, using either red or blue paint. He will decide on the color and pattern for each ball by flipping a fair coin for each of the <math>12</math> decisions he must make. After the paint dries, he will place the <math>6</math> balls in an urn. Frida will randomly select one ball from the urn and note its color and pattern. The events "the ball Frida selects is red" and "the ball Frida selects is striped" may or may not be independent, depending on the outcome of Pablo's coin flips. The probability that these two events are independent can be written as <math>\frac mn,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m?</math> (Recall that two events <math>A</math> and <math>B</math> are independent if <math>P(A \text{ and }B) = P(A) \cdot P(B).</math>) | ||
+ | |||
+ | <math>\textbf{(A) } 243 \qquad \textbf{(B) } 245 \qquad \textbf{(C) } 247 \qquad \textbf{(D) } 249\qquad \textbf{(E) } 251</math> | ||
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[[2024 AMC 12B Problems/Problem 25|Solution]] | [[2024 AMC 12B Problems/Problem 25|Solution]] | ||
==See also== | ==See also== | ||
− | {{AMC12 box|year=2024|ab=B|before=[[ | + | {{AMC12 box|year=2024|ab=B|before=[[2024 AMC 12A Problems]]|after=[[2025 AMC 12A Problems]]}} |
− | + | ||
− | + | [[AMC 12]] | |
− | + | ||
− | + | [[AMC 12 Problems and Solutions]] | |
+ | |||
+ | [[Mathematics competitions]] | ||
+ | |||
+ | [[Mathematics competition resources]] |
Revision as of 00:18, 14 November 2024
2024 AMC 12B (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
[hide]- 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
Problem 2
What is ?
Problem 3
Problem 4
Balls numbered are deposited in bins, labeled and , using the following procedure. Ball is deposited in bin , and balls and are deposited in . The next three balls are deposited in bin , the next in bin , and so on, cycling back to bin after balls are deposited in bin . (For example, are deposited in bin at step 7 of this process.) In which bin is ball deposited?
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
A list of 9 real numbers consists of , , , , , , as well as with . The range of the list is , and the mean and median are both positive integers. How many ordered triples are possible?
Problem 11
Problem 12
Problem 13
Problem 14
How many different remainders can result when the th power of an integer is divided by ?
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Pablo will decorate each of identical white balls with either a striped or a dotted pattern, using either red or blue paint. He will decide on the color and pattern for each ball by flipping a fair coin for each of the decisions he must make. After the paint dries, he will place the balls in an urn. Frida will randomly select one ball from the urn and note its color and pattern. The events "the ball Frida selects is red" and "the ball Frida selects is striped" may or may not be independent, depending on the outcome of Pablo's coin flips. The probability that these two events are independent can be written as where and are relatively prime positive integers. What is (Recall that two events and are independent if )
See also
2024 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by 2024 AMC 12A Problems |
Followed by 2025 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 |