ΒΆ 2022 AMC 10A Problems and Solutions
Problem Set Workbook
Access the downloadable workbook for 2022 AMC 10A problems here.
Discussion Forum
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Individual Problems and Solutions
For problems and detailed solutions to each of the 2022 AMC 10A problems, please refer below:
Problem 1: What is the value of
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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?
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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?
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Problem 4: In some countries, automobile fuel efficiency is measured in liters per kilometers while other countries use miles per gallon. Suppose that 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?
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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 ?
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Problem 6: Which expression is equal to
for ?
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Problem 7: The least common multiple of a positive integer and is , and the greatest common divisor of and is . What is the sum of the digits of ?
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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 possible values of ?
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Problem 9: 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?
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Problem 10: Daniel finds a rectangular index card and measures its diagonal to be centimeters. Daniel then cuts out equal squares of side at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be centimeters, as shown below. What is the area of the original index card?
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Problem 11: Ted mistakenly wrote as . What is the sum of all real numbers for which these two expressions have the same value?
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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?
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Problem 13: Let be a scalene triangle. Point lies on so that bisects . The line through perpendicular to intersects the line through parallel to at point . Suppose and . What is ?
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Problem 14: 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?
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Problem 15: Quadrilateral with side lengths is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form , where , and are positive integers such that and have no common prime factor. What is ?
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Problem 16: The roots of the polynomial are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by units. What is the volume of the new box?
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Problem 17: How many three-digit positive integers are there whose nonzero digits and satisfy
(The bar indicates repetition, thus is the infinite repeating decimal )
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Problem 18: Let be the transformation of the coordinate plane that first rotates the plane degrees counterclockwise around the origin and then reflects the plane across the -axis. What is the least positive integer such that performing the sequence of transformations returns the point back to itself?
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Problem 19: Define as the least common multiple of all the integers from to inclusive. There is a unique integer such that
What is the remainder when is divided by ?
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Problem 20: A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a fourterm geometric sequence of positive integers. The first three terms of the resulting four-term sequence are , and . What is the fourth term of this sequence?
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Problem 21: A bowl is formed by attaching four regular hexagons of side to a square of side . The edges of the adjacent hexagons coincide, as shown in the figure. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl?
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Problem 22: Suppose that cards numbered are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards are picked up on the first pass, and on the second pass, on the third pass, on the fourth pass, and on the fifth pass. For how many of the possible orderings of the cards will the cards be picked up in exactly two passes?
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Problem 23: Isosceles trapezoid has parallel sides and , with and . There is a point in the plane such that , and . What is ?
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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 .)
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Problem 25: Let , and be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the -axis. The left edge of and the right edge of are on the -axis, and contains as many lattice points as does . The top two vertices of are in , and contains of the lattice points contained in . See the figure (not drawn to scale).
The fraction of lattice points in that are in is times the fraction of lattice points in that are in . What is the minimum possible value of the edge length of plus the edge length of plus the edge length of ?
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The problems on this page are the property of the MAA's American Mathematics Competitions