Question bank Math Olympiad Vault 2025

Call a 9-digit number a *cassowary* if it uses each of the digits $1$ through $9$ exactly once. Compute the number of cassowaries that are prime.

Compute
$$
\frac{20 + \frac{1}{25 - \frac{1}{20}}}{25 + \frac{1}{20 - \frac{1}{25}}}.
$$

Compute the largest possible radius of a circle contained in the region defined by $|x + |y|| \leq 1$ in the coordinate plane.

Two rhombi *ABCD* and *AXYZ* are external to each other with a common vertex *A*. The vertices of the rhombi are labelled clockwise. If $\angle DAX = \angle BAZ$, prove that the centres of the rhombi and the midpoint of the segment *BZ* form an isosceles triangle.

Mark writes the expression $\sqrt{abcd}$ on the board, where $abcd$ is a four-digit number and $a \neq 0$. Derek, a toddler, decides to move the $a$, changing Mark's expression to $a\sqrt{bc}$. Surprisingly, these two expressions are equal. Compute the only possible four-digit number $abcd$.

Jacob rolls two fair six-sided dice. If the outcomes of these dice rolls are the same, he rolls a third fair six-sided die. Compute the probability that the sum of outcomes of all the dice he rolls is even.

Jack and Jill play the following game: Jack throws 3 dice and Jill can select some of them, possibly none, and turn each of them to the opposite side. Jill wins if the sum of the values on the dice is a multiple of 4. Can Jill always win? (Note the game is played with standard dice where the sum of the numbers on opposite sides is 7.)

Kelvin the frog is on the bottom-left lily pad of a $3 \times 3$ grid of lily pads, and his home is at the top-right lily pad. He can only jump between two lily pads which are horizontally or vertically adjacent. Compute the number of ways to remove 4 of the lily pads so that the bottom-left and top-right lily pads both remain, but Kelvin cannot get home.

Let $a$, $b$, and $c$ be pairwise distinct positive integers such that $\frac{1}{a}$, $\frac{1}{b}$, $\frac{1}{c}$ is an increasing arithmetic sequence in that order. Prove that $\gcd(a, b) \gt 1$.

Equilateral triangles $\triangle ABC$ and $\triangle DEF$ are drawn such that points $B$, $E$, $F$, and $C$ lie on a line in this order, and point $D$ lies inside triangle $\triangle ABC$. If $BE = 14$, $EF = 15$, and $FC = 16$, compute $AD$.

Compute the sum of the positive divisors (including 1) of $9!$ that have units digit $1$.

Compute the number of ways to arrange the numbers $1$, $2$, $3$, $4$, $5$, $6$, and $7$ around a circle such that the product of every pair of adjacent numbers on the circle is at most $20$. (Rotations and reflections count as different arrangements.)

Ben has 16 balls labeled $1$, $2$, $3$, $\ldots$, $16$, as well as 4 indistinguishable boxes. Two balls are neighbors if their labels differ by $1$. Compute the number of ways for him to put 4 balls in each box such that each ball is in the same box as at least one of its neighbors. (The order in which the balls are placed does not matter.)

Let $\triangle ABC$ be an equilateral triangle with side length $4$. Across all points $P$ inside triangle $\triangle ABC$ satisfying $[PAB] + [PAC] = [PBC]$, compute the minimum possible length of $PA$.

(Here, $[XYZ]$ denotes the area of triangle $\triangle XYZ$.)

Find all 6-digit integers $n$ such that $n$ is a perfect square and that the number formed by the last 3 digits of $n$ is 1 more than the number formed by the first 3 digits of $n$.

In a two-dimensional cave with a parallel floor and ceiling, two stalactites of lengths $16$ and $36$ hang perpendicularly from the ceiling, while two stalagmites of heights $25$ and $49$ grow perpendicularly from the ground. If the tips of these four structures form the vertices of a square in some order, compute the height of the cave.

A polyomino is a connected figure constructed by joining one or more unit squares edge-to-edge. Determine, with proof, the number of non-congruent polyominoes with no holes, perimeter $180$, and area $2024$.

Given that $x$, $y$, and $z$ are positive real numbers such that
$$x^{\log_{2}(yz)} = 2^{8} \cdot 3^{4}, \quad y^{\log_{2}(zx)} = 2^{9} \cdot 3^{6}, \quad \text{and} \quad z^{\log_{2}(xy)} = 2^{5} \cdot 3^{10},$$
compute the smallest possible value of $xyz$.

Point $P$ lies inside square $ABCD$ such that the areas of $\triangle PAB$, $\triangle PBC$, $\triangle PCD$, and $\triangle PDA$ are $1$, $2$, $3$, and $4$, in some order. Compute $PA \cdot PB \cdot PC \cdot PD$.

Let $\lfloor z\rfloor$ denote the greatest integer less than or equal to $z$. Compute
$$
\sum_{j = -1000}^{1000} \left\lfloor \frac{2025}{j + 0.5} \right\rfloor.
$$