Competition · AMC preparation · step 4 of 4

AMC 12 2025A: 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 2025A #1 grade 8+ rate-ratio
Andy and Betsy both live in Mathville. Andy leaves Mathville on his bicycle at 1:30, traveling due north at a steady 8 m…
AMC 12 2025A #2 grade 8+ rate-ratio
A box contains 10 pounds of a nut mix that is 50 percent peanuts, 20 percent cashews, and 30 percent almonds. A second n…
AMC 12 2025A #3 grade 8+ algebra
A team of students is going to compete against a team of teachers in a trivia contest. The total number of students and…
AMC 12 2025A #4 grade 6+ logic
Agnes writes the following four statements on a blank piece of paper. • At least one of these statements is true. • At l…
AMC 12 2025A #5 grade 11+ geometry-2d, algebra
In the figure below, the outside square contains infinitely many squares, each of them with the same center and sides pa…
AMC 12 2025A #6 grade 7+ probability
Six chairs are arranged around a round table. Two students and two teachers randomly select four of the chairs to sit in…
AMC 12 2025A #7 grade 11+ algebra
In a certain alien world, the maximum running speed v of an organism is dependent on its number of toes n and number of…
AMC 12 2025A #8 grade 11+ geometry-2d
Pentagon ABCDE is inscribed in a circle, and ∠ BEC = ∠ CED = 30°. Let line AC and line BD intersect at point F, and supp…
AMC 12 2025A #9 grade 11+ algebra
Let w be the complex number 2+i, where i=√(-1). What real number r has the property that r, w, and w² are three collinea…
AMC 12 2025A #10 grade 10+ geometry-2d
In the figure shown below, major arc AD and minor arc BC have the same center, O. Also, A lies between O and B, and D li…
AMC 12 2025A #11 grade 10+ geometry-2d
The orthocenter of a triangle is the concurrent intersection of the three (possibly extended) altitudes. What is the sum…
AMC 12 2025A #12 grade 11+ algebra
The harmonic mean of a collection of numbers is the reciprocal of the arithmetic mean of the reciprocals of the numbers…
AMC 12 2025A #13 grade 7+ probability, counting
Let C = {1, 2, 3, …, 13}. Let N be the greatest integer such that there exists a subset of C with N elements that does n…
AMC 12 2025A #14 grade 10+ geometry-2d
Points F, G, and H are collinear with G between F and H. The ellipse with foci at G and H is internally tangent to the e…
AMC 12 2025A #15 grade 7+ counting
A set of numbers is called sum-free if whenever x and y are (not necessarily distinct) elements of the set, x+y is not a…
AMC 12 2025A #16 grade 10+ geometry-2d
Triangle △ ABC has side lengths AB = 80, BC = 45, and AC = 75. The bisector of ∠ B and the altitude to side AB intersect…
AMC 12 2025A #17 grade 11+ algebra, geometry-2d
The polynomial (z + i)(z + 2i)(z + 3i) + 10 has three roots in the complex plane, where i = √(-1). What is the area of t…
AMC 12 2025A #18 grade 11+ counting
How many ordered triples (x, y, z) of different positive integers less than or equal to 8 satisfy xy > z, xz > y, and yz…
AMC 12 2025A #19 grade 11+ algebra
Let a, b, and c be the roots of the polynomial x³ + kx + 1. What is the suma³b² + a²b³ + b³c² + b²c³ + c³a² + c²a³?
AMC 12 2025A #20 grade 10+ geometry-3d
The base of the pentahedron shown below is a 13 × 8 rectangle, and its lateral faces are two isosceles triangles with ba…
AMC 12 2025A #21 grade 11+ algebra, number-theory
There is a unique ordered triple (a,k,m) of nonnegative integers such that (4^a + 4^(a+k)+4^(a+2k)+… + 4^(a+mk))/(2^a +…
AMC 12 2025A #22 grade 10+ probability
Three real numbers are chosen independently and uniformly at random between 0 and 1. What is the probability that the gr…
AMC 12 2025A #23 grade 12+ counting
Call a positive integer fair if no digit is used more than once, it has no 0s, and no digit is adjacent to two greater d…
AMC 12 2025A #24 grade 11+ geometry-2d
A circle of radius r is surrounded by 12 circles of radius 1, externally tangent to the central circle and sequentially…
AMC 12 2025A #25 grade 11+ algebra, counting
Polynomials P(x) and Q(x) each have degree 3 and leading coefficient 1, and their roots are all elements of {1,2,3,4,5}.…

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