2025/26 AHSMC Part 1

Two sections of a Math $30$ class take an exam. The maximum possible score is $20$ points. The average score of the $10$ students in the first section is $15$ points. The average score of the $20$ students in the second section is $18$ points. What is the overall average score of all $30$ students combined?

Two solid iron right circular cylinders are each $4$ cm high, and are of diameter $60$ cm and $25$ cm respectively. They are melted down and used to form a single larger right circular cylinder, still $4$ cm high. What is the diameter in cm of the newly created larger cylinder?

I have two six-sided dice: a white one and a red one. Suppose the white one is fair, but the red die is “loaded” such that rolling a “$1$” is twice as likely as rolling any other number. What is the probability that I roll a sum of “$7$” with these two dice?

A rectangular pool table is placed on a grid and has corners at $(0,0)$, $(5,0)$, $(0,3)$, and $(5,3)$. There are pockets only in the four corners. A ball is hit from the corner at $(0,0)$ along the line $y = x$. How many times does the ball bounce off a wall before entering a pocket?

The four distinct digits $1$, $3$, $5$ and $7$ are used to form all possible $4$-digit numbers with no repeated digits. What is the sum of all these possible $4$-digit numbers?

How many integers $N \ge 10$ have the property that removing the last digit of $N$ gives an integer strictly smaller than $\dfrac{N}{11}$?

How many positive integers $n$ satisfy $$n^2 - 15n = 51 - 35\cdot\bigl(2 + (-1)^n\bigr)?$$

For how many integers $n \ge 1$ do there exist real numbers $a$, $b$ such that $$a + b = n \qquad\text{and}\qquad a^2 + b^2 = 25?$$

In a square $ABCD$ with $AB = 1$, let $M$ be the midpoint of side $AB$ and $N$ be a point on side $DC$ such that $DN = 3\,NC$. The segment $AN$ intersects $DM$ at $P$, and the segment $BN$ intersects $CM$ at $Q$.

The area of the quadrilateral $MPNQ$ is

Find the number of pairs $(a,b)$ of integers with $1 < a,b \le 30$ such that $\log_a b$ is an integer.

What is the smallest positive real number $k$ so that the curves $y = x^2 + 10x + 26$ and $y = \sin(kx)$ have at least one point in common? Here, $x$ and $kx$ are in degrees.

A triangular number is any integer of the form $n(n+1)/2$ where $n$ is a positive integer. The number of positive integers $k$ for which $k+1$ is a perfect square and $2k+30$ is a triangular number is

Find the number of pairs $(n,C)$ of positive integers such that $C \le 50$ and the expression $n^2 + C$ is a perfect square.

Let $b$ and $c$ be real numbers, and define the polynomial $P(x) = x^2 + bx + c$. Suppose that $P(P(2)) = P(P(3)) = 0$, and that $P(2) \ne P(3)$. The value of $P(1)$ satisfies

Let $S$ be a set of five different positive integers. The set has a special property: for any non-empty subset of $S$, the average of its elements is an integer. What is the smallest possible value for the largest integer in $S$?

$P$ is a point inside the triangle $\triangle ABC$. We know that $\angle BAP = 30^\circ$, $\angle CAP = 20^\circ$, and $\angle ACP = 50^\circ$ and that the length of the segment $AP$ is equal to the length of the segment $BC$. Find the angle $\angle BCP$.