Competition · AMC preparation · step 4 of 4

AMC 12 2023A: all 25 problems

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

AMC 12 2023A #1 grade 6+ rate-ratio
Cities A and B are 45 miles apart. Alicia lives in A and Beth lives in B. Alicia bikes towards B at 18 miles per hour. L…
AMC 12 2023A #2 grade 7+ arithmetic
The weight of 1/3 of a large pizza together with 3 1/2 cups of orange slices is the same as the weight of 3/4 of a large…
AMC 12 2023A #3 grade 6+ number-theory
How many positive perfect squares less than 2023 are divisible by 5?
AMC 12 2023A #4 grade 8+ arithmetic
How many digits are in the base-ten representation of 8⁵ · 5¹⁰ · 15⁵?
AMC 12 2023A #5 grade 7+ probability
Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability t…
AMC 12 2023A #6 grade 11+ algebra, geometry-2d
Points A and B lie on the graph of y=log₂x. The midpoint of AB is (6, 2). What is the positive difference between the x-…
AMC 12 2023A #7 grade 4+ arithmetic
A digital display shows the current date as an 8-digit integer consisting of a 4-digit year, followed by a 2-digit month…
AMC 12 2023A #8 grade 6+ arithmetic
Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an 11 on the next quiz, her mea…
AMC 12 2023A #9 grade 8+ geometry-2d
A square of area 2 is inscribed in a square of area 3, creating four congruent triangles, as shown below. What is the ra…
AMC 12 2023A #10 grade 8+ algebra
Positive real numbers x and y satisfy y³=x² and (y-x)²=4y². What is x+y?
AMC 12 2023A #11 grade 11+ geometry-2d
What is the degree measure of the acute angle formed by lines with slopes 2 and 1/3?
AMC 12 2023A #12 grade 9+ arithmetic
What is the value of 2³ - 1³ + 4³ - 3³ + 6³ - 5³ + … + 18³ - 17³?
AMC 12 2023A #13 grade 7+ arithmetic
In a table tennis tournament, every participant played every other participant exactly once. Although there were twice a…
AMC 12 2023A #14 grade 12+ algebra
How many complex numbers satisfy the equation z⁵=z, where z is the conjugate of the complex number z?
AMC 12 2023A #15 grade 10+ geometry-2d
Usain is walking for exercise by zigzagging across a 100-meter by 30-meter rectangular field, beginning at point A and e…
AMC 12 2023A #16 grade 11+ algebra
Consider the set of complex numbers z satisfying |1+z+z²|=4. The maximum value of the imaginary part of z can be written…
AMC 12 2023A #17 grade 11+ probability
Flora the frog starts at 0 on the number line and makes a sequence of jumps to the right. In any one jump, independent o…
AMC 12 2023A #18 grade 8+ geometry-2d
Circle C₁ and C₂ each have radius 1, and the distance between their centers is 1/2. Circle C₃ is the largest circle inte…
AMC 12 2023A #19 grade 11+ algebra
What is the product of all solutions to the equation log_7x2023· log_289x2023=log_2023x2023
AMC 12 2023A #20 grade 9+ pattern, number-theory
Rows 1, 2, 3, 4, and 5 of a triangular array of integers are shown below. Each row after the first row is formed by plac…
AMC 12 2023A #21 grade 7+ probability
If A and B are vertices of a polyhedron, define the distance d(A,B) to be the minimum number of edges of the polyhedron…
AMC 12 2023A #22 grade 9+ number-theory
Let f be the unique function defined on the positive integers such that sum_(d| n)d· f(n/d)=1 for all positive integers…
AMC 12 2023A #23 grade 11+ algebra
How many ordered pairs of positive real numbers (a,b) satisfy the equation (1+2a)(2+2b)(2a+b) = 32ab?
AMC 12 2023A #24 grade 10+ counting
Let K be the number of sequences A₁, A₂, …, A_n such that n is a positive integer less than or equal to 10, each A_i is…
AMC 12 2023A #25 grade 12+ algebra
There is a unique sequence of integers a₁, a₂, … a₂₀₂₃ such that tan2023x = (a₁ tan x + a₃ tan³ x + a₅ tan⁵ x + … + a₂₀₂…

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