AMC 10A 2025

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 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?

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

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?

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 $2025$th term in this sequence?

In an equilateral triangle each interior angle is trisected by a pair of rays. The intersection of the interiors of the middle $20^\circ$-angle at each vertex is the interior of a convex hexagon. What is the degree measure of the smallest angle of this hexagon?

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 $b - a$?

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?

Let $f(x) = 100x^3 - 300x^2 + 200x$. For how many real numbers $a$ does the graph of $y = f(x - a)$ pass through the point $(1, 25)$?

A semicircle has diameter $\overline{AB}$ and chord $\overline{CD}$ of length $16$ parallel to $\overline{AB}$. A smaller semicircle with diameter on $\overline{AB}$ and tangent to $\overline{CD}$ is cut from the larger semicircle, as shown. What is the area of the resulting figure, shown shaded?

The sequence $1, x, y, z$ is arithmetic. The sequence $1, p, q, z$ is geometric. Both sequences are strictly increasing and contain only integers, and $z$ is as small as possible. What is the value of $x + y + z + p + q$?

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?

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 $k$, where $0 \lt k \lt 1$. The spaces between squares are alternately shaded as shown in the figure. The area of the shaded portion of the figure is $64\%$ of the area of the original square. What is $k$?

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 sit in two adjacent chairs and the two teachers also sit in two adjacent chairs?

In the figure below, $ABEF$ is a rectangle, $\overline{AD} \perp \overline{DE}$, $AF = 7$, $AB = 1$, and $AD = 5$. What is the area of $\triangle ABC$?

Point $C$ is the intersection of line $\overline{AD}$ extended with side $\overline{BE}$. The figure shows rectangle $ABEF$ with diagonal line from $F$ through $D$ to create point $C$ on $\overline{BE}$.

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 the jar with the most coins?

Let $N$ be the unique positive integer such that dividing $273436$ by $N$ leaves a remainder of $16$ and dividing $272760$ by $N$ leaves a remainder of $15$. What is the tens digit of $N$?

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 $4, 4, 5$ is $\frac{1}{\dfrac{1}{3}\!\left(\dfrac{1}{4}+\dfrac{1}{4}+\dfrac{1}{5}\right)} = \frac{30}{7}.$

What is the harmonic mean of all the real roots of the $4050^{\text{th}}$ degree polynomial $\prod_{k=1}^{2025}(kx^2 - 4x - 3) = (x^2-4x-3)(2x^2-4x-3)(3x^2-4x-3)\cdots(2025x^2-4x-3)\,?$

An array of numbers is constructed beginning with the numbers $-1,\ 3,\ 1$ 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 $-1$ and $1$, respectively.

$\begin{array}{ccccc} -1 & 3 & 1 \\ -1 & 2 & 4 & 1 \\ -1 & 1 & 6 & 5 & 1 \end{array}$

If the process continues, one of the rows will sum to $12{,}288$. In that row, what is the third number from the left?

A silo (right circular cylinder) with diameter $20$ meters stands in a field. MacDonald is located $20$ meters west and $15$ meters south of the center of the silo. McGregor is located $20$ meters east and $g > 0$ meters south of the center of the silo. The line of sight between MacDonald and McGregor is tangent to the silo. The value of $g$ can be written as $\dfrac{a\sqrt{b} - c}{d}$, where $a$, $b$, $c$, and $d$ are positive integers, $b$ is not divisible by the square of any prime, and $d$ is relatively prime to the greatest common divisor of $a$ and $c$. What is $a + b + c + d$?

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 an element of the set. For example, $\{1, 4, 6\}$ and the empty set are sum-free, but $\{2, 4, 5\}$ is not. What is the greatest possible number of elements in a sum-free subset of $\{1, 2, 3, \ldots, 20\}$?

A circle of radius $r$ is surrounded by three circles, whose radii are $1$, $2$, and $3$, all externally tangent to the inner circle and to each other, as shown.

The figure shows four mutually tangent circles: the inner circle of radius $r$, and three outer circles of radii $1$, $2$, and $3$, each pair of outer circles also tangent to each other.

What is $r$?

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

Call a positive integer fair if no digit is used more than once, it has no $0$s, and no digit is adjacent to two greater digits. For example, $196$, $23$, and $12463$ are fair, but $1546$, $320$, and $34321$ are not. How many fair positive integers are there?

A point $P$ is chosen at random inside square $ABCD$. The probability that $\overline{AP}$ is neither the shortest nor the longest side of $\triangle APB$ can be written as $\dfrac{a + b\pi - c\sqrt{d}}{e}$, where $a$, $b$, $c$, $d$, and $e$ are positive integers, $\gcd(a, b, c, e) = 1$, and $d$ is not divisible by the square of any prime. What is $a + b + c + d + e$?