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AMC 10A 2025

Problems, Step-by-Step Hints, and Solutions

AMC 10A/12A 2025 (Problem 1/1)

1.
Andy and Betsy both live in Mathville. Andy leaves Mathville on his bicycle at 1:30 traveling due north at a steady 8 miles per hour. Betsy leaves on her bicycle from the same point at 2:30, traveling due east at a steady 12 miles per hour. At what time will they be exactly the same distance from their common starting point?
(A)  3:30\text{(A)}\;3:30(B)  3:45\text{(B)}\;3:45(C)  4:00\text{(C)}\;4:00(D)  4:15\text{(D)}\;4:15(E)  4:30\text{(E)}\;4:30

AMC 10A/12A 2025 (Problem 2/2)

2.
A box contains 10 pounds of a nut mix that is 50 percent peanuts, 20 percent cashews, and 30 percent almonds. A second nut mix containing 20 percent peanuts, 40 percent cashews, and 40 percent almonds is added to the box resulting in a new nut mix that is 40 percent peanuts. How many pounds of cashews are now in the box?
(A)  3.5\text{(A)}\;3.5(B)  4\text{(B)}\;4(C)  4.5\text{(C)}\;4.5(D)  5\text{(D)}\;5(E)  6\text{(E)}\;6

AMC 10A 2025 (Problem 3)

3.
How many isosceles triangles are there with positive area whose side lengths are all positive integers and whose longest side has length 2025?
(A)  2025\text{(A)}\;2025(B)  2026\text{(B)}\;2026(C)  3012\text{(C)}\;3012(D)  3037\text{(D)}\;3037(E)  4050\text{(E)}\;4050

AMC 10A/12A 2025 (Problem 4/3)

4.
A team of students is going to compete against a team of teachers in a trivia contest. The total number of students and teachers is 15. Ash, a cousin of one of the students, wants to join the contest. If Ash plays with the students, the average age on that team will increase from 12 to 14. If Ash plays with the teachers, the average age on that team will decrease from 55 to 52. How old is Ash?
(A)  28\text{(A)}\;28(B)  29\text{(B)}\;29(C)  30\text{(C)}\;30(D)  32\text{(D)}\;32(E)  33\text{(E)}\;33

AMC 10A 2025 (Problem 5)

5.
Consider the sequence of positive integers1,2,1,2,3,2,1,2,3,4,3,2,1,2,3,4,5,4,3,2,1,…1,2,1,2,3,2,1,2,3,4,3,2,1,2,3,4,5,4,3,2,1,\ldots
What is the 2025th term in this sequence?
(A)  5\text{(A)}\;5(B)  15\text{(B)}\;15(C)  16\text{(C)}\;16(D)  44\text{(D)}\;44(E)  45\text{(E)}\;45

AMC 10A 2025 (Problem 6)

6.
In an equilateral triangle each interior angle is trisected by a pair of rays. The intersection of the interiors of the middle 20°-angle at each vertex is the interior of a convex hexagon. What is the degree measure of the smallest angle of this hexagon?
(A)  80\text{(A)}\;80(B)  90\text{(B)}\;90(C)  100\text{(C)}\;100(D)  110\text{(D)}\;110(E)  120\text{(E)}\;120

AMC 10A 2025 (Problem 7)

7.
Suppose aa and bb are real numbers. When the polynomial x3+x2+ax+bx^3+x^2+ax+b is divided by x−1x-1, the remainder is 44. When the polynomial is divided by x−2x-2, the remainder is 66. What is b−ab-a?
(A)  14\text{(A)}\;14(B)  15\text{(B)}\;15(C)  16\text{(C)}\;16(D)  17\text{(D)}\;17(E)  18\text{(E)}\;18

AMC 10A/12A 2025 (Problem 8/4)

8.
Agnes writes the following four statements on a blank piece of paper.

• At least one of these statements is true.
• At least two of these statements are true.
• At least two of these statements are false.
• At least one of these statements is false.

Each statement is either true or false. How many false statements did Agnes write on the paper?
(A)  0\text{(A)}\;0(B)  1\text{(B)}\;1(C)  2\text{(C)}\;2(D)  3\text{(D)}\;3(E)  4\text{(E)}\;4

AMC 10A 2025 (Problem 9)

9.
Let f(x)=100x3−300x2+200xf(x)=100x^3-300x^2+200x. For how many real numbers aa does the graph of y=f(x−a)y=f(x-a) pass through the point (1,25)(1,25)?
(A)  1\text{(A)}\;1(B)  2\text{(B)}\;2(C)  3\text{(C)}\;3(D)  4\text{(D)}\;4(E)  more than 4\text{(E)}\;\text{more than 4}

AMC 10A 2025 (Problem 10)

10.
A semicircle has diameter AB‾\overline{AB} and chord CD‾\overline{CD} of length 1616 parallel to AB‾\overline{AB}. A smaller semicircle with diameter on AB‾\overline{AB} and tangent to CD‾\overline{CD} is cut from the larger semicircle, as shown below. What is the area of the resulting figure, shown shaded?
Semicircle with diameter AB and chord CD
(A)  16π\text{(A)}\;16\pi(B)  24π\text{(B)}\;24\pi(C)  32π\text{(C)}\;32\pi(D)  48π\text{(D)}\;48\pi(E)  64π\text{(E)}\;64\pi

AMC 10A 2025 (Problem 11)

11.
The sequence 1,x,y,z1,x,y,z is arithmetic. The sequence 1,p,q,z1,p,q,z is geometric. Both sequences are strictly increasing and contain only integers, and zz is as small as possible. What is the value of x+y+z+p+qx+y+z+p+q?
(A)  66\text{(A)}\;66(B)  91\text{(B)}\;91(C)  103\text{(C)}\;103(D)  132\text{(D)}\;132(E)  149\text{(E)}\;149

AMC 10A 2025 (Problem 12)

12.
Carlos uses a 4-digit passcode to unlock his computer. In his passcode, exactly one digit is even, exactly one (possibly different) digit is prime, and no digit is 0. How many 4-digit passcodes satisfy these conditions?
(A)  176\text{(A)}\;176(B)  192\text{(B)}\;192(C)  432\text{(C)}\;432(D)  464\text{(D)}\;464(E)  608\text{(E)}\;608

AMC 10A/12A 2025 (Problem 13/5)

13.
In the figure below, the outside square contains infinitely many squares, each of them with the same center and sides parallel to the outside square. The ratio of the side length of a square to the side length of the next inner square is kk, where 0<k<10<k<1. The spaces between squares are alternately shaded as shown in the figure (which is not necessarily drawn to scale).

The area of the shaded portion of the figure is 64%64\% of the area of the original square. What is kk?
Nested squares with alternating shaded regions
(A)  35\text{(A)}\;\frac{3}{5}(B)  1625\text{(B)}\;\frac{16}{25}(C)  23\text{(C)}\;\frac{2}{3}(D)  34\text{(D)}\;\frac{3}{4}(E)  45\text{(E)}\;\frac{4}{5}

AMC 10A/12A 2025 (Problem 14/6)

14.
Six chairs are arranged around a round table. Two students and two teachers randomly select four of the chairs to sit in. What is the probability that the two students will sit in two adjacent chairs and the two teachers will also sit in two adjacent chairs?
(A)  16\text{(A)}\;\frac{1}{6}(B)  15\text{(B)}\;\frac{1}{5}(C)  29\text{(C)}\;\frac{2}{9}(D)  313\text{(D)}\;\frac{3}{13}(E)  14\text{(E)}\;\frac{1}{4}

AMC 10A 2025 (Problem 15)

15.
In the figure below, ABEFABEF is a rectangle, AD‾⊥DE‾\overline{AD}\perp\overline{DE}, AF=7AF=7, AB=1AB=1, and AD=5AD=5. What is the area of △ABC\triangle ABC?
Rectangle ABEF with triangle ABC
(A)  38\text{(A)}\;\frac38(B)  49\text{(B)}\;\frac49(C)  1813\text{(C)}\;\frac18\sqrt{13}(D)  715\text{(D)}\;\frac7{15}(E)  1815\text{(E)}\;\frac18\sqrt{15}

AMC 10A 2025 (Problem 16)

16.
There are three jars. Each of three coins is placed in one of the three jars, chosen at random and independently of the placements of the other coins. What is the expected number of coins in a jar with the most coins?
(A)  43\text{(A)}\;\frac{4}{3}(B)  139\text{(B)}\;\frac{13}{9}(C)  53\text{(C)}\;\frac{5}{3}(D)  179\text{(D)}\;\frac{17}{9}(E)  2\text{(E)}\;2

AMC 10A 2025 (Problem 17)

17.
Let NN be the unique positive integer such that dividing 273436273436 by NN leaves a remainder of 1616 and dividing 272760272760 by NN leaves a remainder of 1515. What is the tens digit of NN?
(A)  0\text{(A)}\;0(B)  1\text{(B)}\;1(C)  2\text{(C)}\;2(D)  3\text{(D)}\;3(E)  4\text{(E)}\;4

AMC 10A/12A 2025 (Problem 18/12)

18.
The harmonic mean of a collection of numbers is the reciprocal of the arithmetic mean of the reciprocals of the numbers in the collection. For example, the harmonic mean of 4,4,4,4, and 55 is 113(14+14+15)=307\frac{1}{\frac13\left(\frac14+\frac14+\frac15\right)} = \frac{30}{7} What is the harmonic mean of all the real roots of the 4050th-degree polynomial ∏k=12025(kx2−4x−3)=(x2−4x−3)(2x2−4x−3)(3x2−4x−3)⋯(2025x2−4x−3)?\prod_{k=1}^{2025} \left(kx^2-4x-3\right) = (x^2-4x-3)(2x^2-4x-3)(3x^2-4x-3)\cdots(2025x^2-4x-3)?
(A)  −53\text{(A)}\;-\frac{5}{3}(B)  −32\text{(B)}\;-\frac{3}{2}(C)  −65\text{(C)}\;-\frac{6}{5}(D)  −56\text{(D)}\;-\frac{5}{6}(E)  −23\text{(E)}\;-\frac{2}{3}

AMC 10A 2025 (Problem 19)

19.
An array of numbers is constructed beginning with the numbers −1-1, 33, and 11 in the top row. Each adjacent pair of numbers is summed to produce a number in the next row. Each row begins and ends with −1-1 and 11, respectively. If the process continues, one of the rows will sum to 12,28812,288. In that row, what is the third number from the left?
AMC 10A 2025 Problem 19 diagram
(A)  −29\text{(A)}\;-29(B)  −21\text{(B)}\;-21(C)  −14\text{(C)}\;-14(D)  −8\text{(D)}\;-8(E)  −3\text{(E)}\;-3

AMC 10A 2025 (Problem 20)

20.
A silo (right circular cylinder) with diameter 2020 meters stands in a field. MacDonald is located 2020 meters west and 1515 meters south of the center of the silo. McGregor is located 2020 meters east and g>0g>0 meters south of the center of the silo. The line of sight between MacDonald and McGregor is tangent to the silo. The value of gg can be written as ab−cd\frac{a\sqrt{b}-c}{d}, where aa, bb, cc, and dd are positive integers, bb is not divisible by the square of any prime, and dd is relatively prime to the greatest common divisor of aa and cc. What is a+b+c+da+b+c+d?
(A)  119\text{(A)}\;119(B)  120\text{(B)}\;120(C)  121\text{(C)}\;121(D)  122\text{(D)}\;122(E)  123\text{(E)}\;123

AMC 10A/12A 2025 (Problem 21/15)

21.
A set of numbers is called sum-free if whenever xx and yy are (not necessarily distinct) elements of the set, x+yx+y is not an element of the set. For example, {1,4,6}\{1,4,6\} and the empty set are sum-free, but {2,4,5}\{2,4,5\} is not. What is the greatest possible number of elements in a sum-free subset of {1,2,3,…,20}\{1,2,3,\ldots,20\}?
(A)  8\text{(A)}\;8(B)  9\text{(B)}\;9(C)  10\text{(C)}\;10(D)  11\text{(D)}\;11(E)  12\text{(E)}\;12

AMC 10A 2025 (Problem 22)

22.
A circle of radius rr is surrounded by three circles, whose radii are 11, 22, and 33, all externally tangent to the inner circle and externally tangent to each other, as shown below. What is rr?
Three circles tangent to each other and the inner circle
(A)  14\text{(A)}\;\frac{1}{4}(B)  623\text{(B)}\;\frac{6}{23}(C)  311\text{(C)}\;\frac{3}{11}(D)  517\text{(D)}\;\frac{5}{17}(E)  310\text{(E)}\;\frac{3}{10}

AMC 10A/12A 2025 (Problem 23/16)

23.
Triangle ABCABC has side lengths AB=80AB=80, BC=45BC=45, and AC=75AC=75. The bisector of ∠B\angle B and the altitude to side ABAB intersect at point PP. What is BPBP?
(A)  18\text{(A)}\;18(B)  19\text{(B)}\;19(C)  20\text{(C)}\;20(D)  21\text{(D)}\;21(E)  22\text{(E)}\;22

AMC 10A/12A 2025 (Problem 24/23)

24.
Call a positive integer fair if no digit is used more than once, it has no 00s, and no digit is adjacent to two greater digits. For example, 196196, 2323, and 1246312463 are fair, but 15461546, 320320, and 3432134321 are not fair. How many four-digit positive integers are fair?
(A)  511\text{(A)}\;511(B)  2584\text{(B)}\;2584(C)  9841\text{(C)}\;9841(D)  17711\text{(D)}\;17711(E)  19682\text{(E)}\;19682

AMC 10A 2025 (Problem 25)

25.
A point PP is chosen at random inside square ABCDABCD. The probability that AP‾\overline{AP} is neither the shortest nor the longest side of △APB\triangle APB can be written as a+bπ−cde\frac{a+b\pi-c\sqrt{d}}{e}, where a,b,c,d,ea,b,c,d,e are positive integers, gcd⁡(a,b,c,e)=1\gcd(a,b,c,e)=1, and dd is not divisible by the square of a prime. What is a+b+c+d+ea+b+c+d+e?
(A)  25\text{(A)}\;25(B)  26\text{(B)}\;26(C)  27\text{(C)}\;27(D)  28\text{(D)}\;28(E)  29\text{(E)}\;29