Difference between revisions of "2022 AMC 10A Problems"
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==Problem 12== | ==Problem 12== | ||
− | + | On Halloween <math>31</math> children walked into the principal's office asking for candy. They | |
+ | can be classified into three types: Some always lie; some always tell the truth; and | ||
+ | some alternately lie and tell the truth. The alternaters arbitrarily choose their first | ||
+ | response, either a lie or the truth, but each subsequent statement has the opposite | ||
+ | truth value from its predecessor. The principal asked everyone the same three | ||
+ | questions in this order. | ||
− | <math>\textbf{(A) } | + | "Are you a truth-teller?" The principal gave a piece of candy to each of the <math>22</math> |
+ | children who answered yes. | ||
+ | |||
+ | "Are you an alternater?" The principal gave a piece of candy to each of the <math>15</math> | ||
+ | children who answered yes. | ||
+ | |||
+ | "Are you a liar?" The principal gave a piece of candy to each of the <math>9</math> children who | ||
+ | answered yes. | ||
+ | |||
+ | How many pieces of candy in all did the principal give to the children who always | ||
+ | tell the truth? | ||
+ | |||
+ | <math>\textbf{(A) } 7 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 21 \qquad \textbf{(D) } 27 \qquad \textbf{(E) } 31</math> | ||
[[2022 AMC 10A Problems/Problem 12|Solution]] | [[2022 AMC 10A Problems/Problem 12|Solution]] |
Revision as of 20:54, 11 November 2022
2022 AMC 10A (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
What is the value of
Problem 2
Mike cycled laps in minutes. Assume he cycled at a constant speed throughout. Approximately how many laps did he complete in the first minutes?
Problem 3
The sum of three numbers is The first number is times the third number, and the third number is less than the second number. What is the absolute value of the difference between the first and second numbers?
Problem 4
In some countries, automobile fuel efficiency is measured in liters per kilometers while other countries use miles per gallon. Suppose that 1 kilometer equals miles, and gallon equals liters. Which of the following gives the fuel efficiency in liters per kilometers for a car that gets miles per gallon?
Problem 5
Square has side length . Points , , , and each lie on a side of such that is an equilateral convex hexagon with side length . What is ?
Problem 6
Which expression is equal to for
Problem 7
The least common multiple of a positive divisor and is , and the greatest common divisor of and is . What is the sum of the digits of ?
Problem 8
A data set consists of not distinct) positive integers: , , , , , and . The average (arithmetic mean) of the numbers equals a value in the data set. What is the sum of all positive values of ?
Problem 9
XXX
Problem 10
XXX
Problem 11
XXX
Problem 12
On Halloween children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order.
"Are you a truth-teller?" The principal gave a piece of candy to each of the children who answered yes.
"Are you an alternater?" The principal gave a piece of candy to each of the children who answered yes.
"Are you a liar?" The principal gave a piece of candy to each of the children who answered yes.
How many pieces of candy in all did the principal give to the children who always tell the truth?
Problem 13
XXX
Problem 14
XXX
Problem 15
XXX
Problem 16
XXX
Problem 17
XXX
Problem 18
XXX
Problem 19
XXX
Problem 20
XXX
Problem 21
XXX
Problem 22
XXX
Problem 23
XXX
Problem 24
XXX
Problem 25
XXX
See also
2020 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by 2021 Fall AMC 10B Problems |
Followed by 2022 AMC 10B 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.