Competition · AMC preparation · step 4 of 4

AMC 12 2024A: 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 2024A #1 grade 7+ arithmetic, algebra
What is the value of 9901·101-99·10101?
AMC 12 2024A #2 grade 8+ algebra, rate-ratio
A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form T=aL+bG, whe…
AMC 12 2024A #3 grade 6+ arithmetic, number-theory
The number 2024 is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-dig…
AMC 12 2024A #4 grade 6+ number-theory
What is the least value of n such that n! is a multiple of 2024?
AMC 12 2024A #5 grade 7+ arithmetic, algebra
A data set containing 20 numbers, some of which are 6, has mean 45. When all the 6s are removed, the data set has mean 6…
AMC 12 2024A #6 grade 7+ number-theory, arithmetic
The product of three integers is 60. What is the least possible positive sum of the three integers?
AMC 12 2024A #7 grade 12+ geometry-2d
In Δ ABC, ∠ ABC = 90° and BA = BC = √2. Points P₁, P₂, …, P₂₀₂₄ lie on hypotenuse AC so that AP₁= P₁P₂ = P₂P₃ = … = P₂₀₂…
AMC 12 2024A #8 grade 11+ algebra
How many angles θ with 0≤θ≤2π satisfy log(sin(3θ))+log(cos(2θ))=0?
AMC 12 2024A #9 grade 8+ number-theory, algebra
Let M be the greatest integer such that both M+1213 and M+3773 are perfect squares. What is the units digit of M?
AMC 12 2024A #10 grade 11+ geometry-2d
Let α be the radian measure of the smallest angle in a 3-4-5 right triangle. Let β be the radian measure of the smallest…
AMC 12 2024A #11 grade 8+ number-theory
There are exactly K positive integers b with 5 ≤ b ≤ 2024 such that the base-b integer 2024_b is divisible by 16 (where…
AMC 12 2024A #12 grade 6+ algebra, number-theory
The first three terms of a geometric sequence are the integers a, 720, and b, where a<720<b. What is the sum of the digi…
AMC 12 2024A #13 grade 11+ algebra
The graph of y=e^(x+1)+e^(-x)-2 has an axis of symmetry. What is the reflection of the point (-1,1/2) over this axis?
AMC 12 2024A #14 grade 8+ algebra
The numbers, in order, of each row and the numbers, in order, of each column of a 5 × 5 array of integers form an arithm…
AMC 12 2024A #15 grade 11+ algebra
AMC 12 2024A #16 grade 11+ probability, counting
A set of 12 tokens — 3 red, 2 white, 1 blue, and 6 black — is to be distributed at random to 3 game players, 4 tokens pe…
AMC 12 2024A #17 grade 8+ algebra, number-theory
Integers a, b, and c satisfy ab + c = 100, bc + a = 87, and ca + b = 60. What is ab + bc + ca?
AMC 12 2024A #18 grade 10+ geometry-2d
On top of a rectangular card with sides of length 1 and 2+√3, an identical card is placed so that two of their diagonals…
AMC 12 2024A #19 grade 11+ geometry-2d
Cyclic quadrilateral ABCD has lengths BC=CD=3 and DA=5 with ∠ CDA=120°. What is the length of the shorter diagonal of AB…
AMC 12 2024A #20 grade 10+ probability, geometry-2d
Points P and Q are chosen uniformly and independently at random on sides AB and AC, respectively, of equilateral triangl…
AMC 12 2024A #21 grade 9+ algebra
Suppose that a₁ = 2 and the sequence (a_n) satisfies the recurrence relation (a_n -1)/(n-1)=(a_(n-1)+1)/((n-1)+1)for all…
AMC 12 2024A #22 grade 7+ counting, geometry-2d
The figure below shows a dotted grid 8 cells wide and 3 cells tall consisting of 1''×1'' squares. Carl places 1-inch too…
AMC 12 2024A #23 grade 12+ algebra
What is the value of tan² π/16 · tan² 3π/16 + tan² π/16 · tan² 5π/16+tan² 3π/16 · tan² 7π/16+tan² 5π/16 · tan² 7π/16?
AMC 12 2024A #24 grade 11+ geometry-3d, number-theory
A disphenoid is a tetrahedron whose triangular faces are congruent to one another. What is the least total surface area…
AMC 12 2024A #25 grade 11+ counting, algebra
A graph is symmetric about a line if the graph remains unchanged after reflection in that line. For how many quadruples…

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