Competition · AMC preparation · step 4 of 4

AMC 10 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 10 2025A #1 grade 6+ 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 10 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 10 2025A #3 grade 7+ geometry-2d
How many isosceles triangles are there with positive area whose side lengths are all positive integers and whose longest…
AMC 10 2025A #4 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 10 2025A #5 grade 5+ pattern
Consider the sequence of positive integers 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 4, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6,…
AMC 10 2025A #6 grade 8+ geometry-2d
In an equilateral triangle each interior angle is trisected by a pair of rays. The intersection of the interiors of the…
AMC 10 2025A #7 grade 8+ algebra
Suppose a and b are real numbers. When the polynomial x³+x²+ax+b is divided by x-1, the remainder is 4. When the polynom…
AMC 10 2025A #8 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 10 2025A #9 grade 8+ algebra
Let f(x) = 100x³ - 300x² + 200x. For how many real numbers a does the graph of y = f(x - a) pass through the point (1, 2…
AMC 10 2025A #10 grade 8+ geometry-2d
A semicircle has diameter AB and chord CD of length 16 parallel to AB. A smaller semicircle with diameter on AB and tang…
AMC 10 2025A #11 grade 6+ algebra
The sequence 1,x,y,z is arithmetic. The sequence 1,p,q,z is geometric. Both sequences are strictly increasing and contai…
AMC 10 2025A #12 grade 7+ counting
Carlos uses a 4-digit passcode to unlock his computer. In his passcode, exactly one digit is even, exactly one (possibly…
AMC 10 2025A #13 grade 8+ geometry-2d
In the figure below, the outside square contains infinitely many squares, each of them with the same center and sides pa…
AMC 10 2025A #14 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 10 2025A #15 grade 8+ geometry-2d
In the figure below, ABEF is a rectangle, AD⊥DE, AF=7, AB=1, and AD=5. What is the area of △ ABC?
AMC 10 2025A #16 grade 7+ probability
There are three jars. Each of three coins is placed in one of the three jars, chosen at random and independently of the…
AMC 10 2025A #17 grade 6+ number-theory
Let N be the unique positive integer such that dividing 273436 by N leaves a remainder of 16 and dividing 272760 by N le…
AMC 10 2025A #18 grade 7+ algebra
The harmonic mean of a collection of numbers is the reciprocal of the arithmetic mean of the reciprocals of the numbers…
AMC 10 2025A #19 grade 7+ arithmetic
An array of numbers is constructed beginning with the numbers -1, 3, and 1 in the top row. Each adjacent pair of numbers…
AMC 10 2025A #20 grade 8+ geometry-2d
A silo (right circular cylinder) with diameter 20 meters stands in a field. MacDonald is located 20 meters west and 15 m…
AMC 10 2025A #21 grade 3+ number-theory
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 10 2025A #22 grade 8+ geometry-2d
A circle of radius r is surrounded by three circles, whose radii are 1, 2, and 3, all externally tangent to the inner ci…
AMC 10 2025A #23 grade 8+ 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 10 2025A #24 grade 8+ 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 10 2025A #25 grade 8+ geometry-2d
A point P is chosen at random inside square ABCD. The probability that AP is neither the shortest nor the longest side o…

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