Difference between revisions of "2022 AMC 12A Problems"
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− | + | Let <math>h_n</math> and <math>k_n</math> be the unique relatively prime positive integers such that <cmath>\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...\frac{1}{n}=\frac{h_n}{k_n}.</cmath> Let <math>L_n</math> denote the least common multiple of the numbers <math>1, 2, 3, ..., n</math>. For how many integers with <math>1\le{n}\le{22}</math> is <math>k_n<L_n</math>? | |
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[[2022 AMC 12A Problems/Problem 23|Solution]] | [[2022 AMC 12A Problems/Problem 23|Solution]] |
Revision as of 11:17, 12 November 2022
2022 AMC 12A (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
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 3
These problems will be posted once the 2022 AMC 12A is released.
Problem 4
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 5
The between points and in the coordinate plane is given by . For how many points with integer coordinates is the taxicab distance between and the origin less than or equal to ?
Problem 6
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 7
A rectangle is partitioned into regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?
Problem 8
The infinite product evaluates to a real number. What is that number?
Problem 9
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 10
How many ways are there to split the integers through into pairs such that in each pair, the greater number is at least times the lesser number?
Problem 11
What is the product of all real numbers such that the distance on the number line between and is twice the distance on the number line between and ?
Problem 12
Let be the midpoint of in regular tetrahedron . What is ?
Problem 13
Let be the region in the complex plane consisting of all complex numbers that can be written as the sum of complex numbers and , where lies on the segment with endpoints and , and has magnitude at most . What integer is closest to the area of ?
Problem 14
These problems will be posted once the 2022 AMC 12A is released.
Problem 15
These problems will be posted once the 2022 AMC 12A is released.
Problem 16
These problems will be posted once the 2022 AMC 12A is released.
Problem 17
These problems will be posted once the 2022 AMC 12A is released.
Problem 18
These problems will be posted once the 2022 AMC 12A is released.
Problem 19
These problems will be posted once the 2022 AMC 12A is released.
Problem 20
These problems will be posted once the 2022 AMC 12A is released.
Problem 21
These problems will be posted once the 2022 AMC 12A is released.
Problem 22
These problems will be posted once the 2022 AMC 12A is released.
Problem 23
Let and be the unique relatively prime positive integers such that Let denote the least common multiple of the numbers . For how many integers with is ?
Problem 24
How many strings of length formed from the digits , , , , are there such that for each , at least of the digits are less than ? (For example, satisfies this condition because it contains at least digit less than , at least digits less than , at least digits less than , and at least digits less than . The string does not satisfy the condition because it does not contain at least digits less than .)
Problem 25
A circle with integer radius is centered at . Distinct line segments of length connect points to for and are tangent to the circle, where , , and are all positive integers and . What is the ratio for the least possible value of ?
See also
2022 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by 2021 AMC 12B Problems |
Followed by 2022 AMC 12B 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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.