Competition · AMC preparation · step 4 of 4

AMC 10 2022A: all 25 problems

Each problem has a solution worked from first principles and the first grade (by CCSS standards) that can solve it.

AMC 10 2022A #1 grade 5+ arithmetic
What is the value of 3+1/(3+1/(3+1/3))?
AMC 10 2022A #2 grade 6+ rate-ratio
Mike cycled 15 laps in 57 minutes. Assume he cycled at a constant speed throughout. Approximately how many laps did he c…
AMC 10 2022A #3 grade 6+ arithmetic
The sum of three numbers is 96. The first number is 6 times the third number, and the third number is 40 less than the s…
AMC 10 2022A #4 grade 6+ arithmetic
In some countries, automobile fuel efficiency is measured in liters per 100 kilometers while other countries use miles p…
AMC 10 2022A #5 grade 8+ geometry-2d
Square ABCD has side length 1. Points P, Q, R, and S each lie on a side of ABCD such that APQCRS is an equilateral conve…
AMC 10 2022A #6 grade 8+ algebra
Which expression is equal to |a-2-√((a-1)²)| for a<0?
AMC 10 2022A #7 grade 6+ number-theory
The least common multiple of a positive integer n and 18 is 180, and the greatest common divisor of n and 45 is 15. What…
AMC 10 2022A #8 grade 6+ arithmetic
A data set consists of 6 (not distinct) positive integers: 1, 7, 5, 2, 5, and X. The average (arithmetic mean) of the 6…
AMC 10 2022A #9 grade 7+ geometry-2d
A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color - red, orange, yellow, bl…
AMC 10 2022A #10 grade 8+ geometry-2d
Daniel finds a rectangular index card and measures its diagonal to be 8 centimeters. Daniel then cuts out equal squares…
AMC 10 2022A #11 grade 8+ arithmetic
Ted mistakenly wrote 2^m·√(1/4096) as 2·^m√(1/4096). What is the sum of all real numbers m for which these two expressio…
AMC 10 2022A #12 grade 6+ arithmetic
On Halloween 31 children walked into the principal's office asking for candy. They can be classified into three types: S…
AMC 10 2022A #13 grade 8+ geometry-2d
Let △ ABC be a scalene triangle. Point P lies on BC so that AP bisects ∠ BAC. The line through B perpendicular to AP int…
AMC 10 2022A #14 grade 7+ counting
How many ways are there to split the integers 1 through 14 into 7 pairs such that in each pair, the greater number is at…
AMC 10 2022A #15 grade 8+ geometry-2d
Quadrilateral ABCD with side lengths AB=7, BC=24, CD=20, DA=15 is inscribed in a circle. The area interior to the circle…
AMC 10 2022A #16 grade 8+ geometry-3d
The roots of the polynomial 10x³ - 39x² + 29x - 6 are the height, length, and width of a rectangular box (right rectangu…
AMC 10 2022A #17 grade 7+ arithmetic
How many three-digit positive integers a b c are there whose nonzero digits a,b, and c satisfy 0.a b c = 1/3 (0.a + 0.b…
AMC 10 2022A #18 grade 8+ arithmetic
Let T_k be the transformation of the coordinate plane that first rotates the plane k degrees counterclockwise around the…
AMC 10 2022A #19 grade 7+ number-theory
Define L_n as the least common multiple of all the integers from 1 to n inclusive. There is a unique integer h such that…
AMC 10 2022A #20 grade 8+ arithmetic
A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corres…
AMC 10 2022A #21 grade 8+ geometry-2d
A bowl is formed by attaching four regular hexagons of side 1 to a square of side 1. The edges of the adjacent hexagons…
AMC 10 2022A #22 grade 8+ arithmetic
Suppose that 13 cards numbered 1, 2, 3, …, 13 are arranged in a row. The task is to pick them up in numerically increasi…
AMC 10 2022A #23 grade 8+ geometry-2d
Isosceles trapezoid ABCD has parallel sides AD and BC, with BC < AD and AB = CD. There is a point P in the plane such th…
AMC 10 2022A #24 grade 6+ arithmetic
How many strings of length 5 formed from the digits 0, 1, 2, 3, 4 are there such that for each j ∈ {1,2,3,4}, at least j…
AMC 10 2022A #25 grade 8+ geometry-2d
Let R, S, and T be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in th…

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