2025 AMC 10A Problems: Difference between revisions

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2025 AMC 10A (Answer Key) Printable versions: Wiki • AoPS Resources • PDF
Instructions This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the SAT if before 2006. No problems on the test will require the use of a calculator). Figures are not necessarily drawn to scale. You will have 75 minutes working time to complete the test.
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Problem 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?

$\textbf{(A) } 3{:}30 \qquad\textbf{(B) } 3{:}45 \qquad\textbf{(C) } 4{:}00 \qquad\textbf{(D) } 4{:}15 \qquad\textbf{(E) } 4{:}30$

Problem 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?

$\textbf{(A) } 3.5 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 4.5 \qquad\textbf{(D) } 5 \qquad\textbf{(E) } 6$

Problem 3

How many isosceles triangles are there with positive area whose side lengths are all positive integers and whose longest side has length 2025?

$\textbf{(A) } 2025 \qquad\textbf{(B) } 2026 \qquad\textbf{(C) } 3012 \qquad\textbf{(D) } 3037 \qquad\textbf{(E) } 4050$

Problem 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?

$\textbf{(A) } 28 \qquad\textbf{(B) } 29 \qquad\textbf{(C) } 30 \qquad\textbf{(D) } 32 \qquad\textbf{(E) } 33$

Problem 5

Consider the sequence of positive integers

$1,2,1,2,3,2,1,2,3,4,3,2,1,2,3,4,5,4,3,2,1,2,3,4,5,6,5,4,3,2,1,2,\dots$

What is the 2025th term in this sequence?

$\textbf{(A) } 5 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 16 \qquad\textbf{(D) } 44 \qquad\textbf{(E) } 45$

Problem 6

Problem 7

Suppose $a$ and $b$ are real numbers. When the polynomial $x^3+x^2+ax+b$ is divided by $x-1$ , the remainder is $4$ . When the polynomial is divided by $x-2$ , the remainder is $6$ . What is $b-a$ ?

$\textbf{(A) } 14 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 16 \qquad\textbf{(D) } 17 \qquad\textbf{(E) } 18$

Solution

Problem 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?

Answer Choices:

A. 0

B. 1

C. 2

D. 3

E. 4

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 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 two students will sit in two adjacent chairs and the two teachers will sit in two adjacent chairs? Correct ans: 1/5

Solution: 6 * 3 * 2 factorial squared/(6 choose 4 * 4!) = 1/5

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

This is a problem that's easier than problem 1 but being put to problem 22 to mess up our points

Problem 23

Triangle $\triangle ABC$ has side lengths $AB = 80$ , $BC = 45$ , and $AC = 75$ . The bisector $\angle B$ and the altitude to side $\overline{AB}$ intersect at point $P$ . What is $BP$ ?

$\textbf{(A)}~18\qquad\textbf{(B)}~19\qquad\textbf{(C)}~20\qquad\textbf{(D)}~21\qquad\textbf{(E)}~22$

@@ Line 24: / Line 24: @@
 ==Problem 5==
+Consider the sequence of positive integers
+<cmath> 1,2,1,2,3,2,1,2,3,4,3,2,1,2,3,4,5,4,3,2,1,2,3,4,5,6,5,4,3,2,1,2,\dots </cmath>
+What is the 2025th term in this sequence?
+<imath>\textbf{(A) } 5 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 16 \qquad\textbf{(D) } 44 \qquad\textbf{(E) } 45</imath>
 ==Problem 6==

2025 AMC 10A (Problems • Answer Key • Resources)
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