We are given a triangle $ABC$ such that $\angle BAC < 90^{\circ}$. The point $D$ is on the opposite side of the line $AB$ to $C$ such that $|AD| = |BD|$ and $\angle ADB = 90^{\circ}$. Similarly, the point $E$ is on the opposite side of $AC$ to $B$ such that $|AE| = |CE|$ and $\angle AEC = 90^{\circ}$. The point $X$ is such that $ADXE$ is a parallelogram. Prove that $|BX| = |CX|$.

For $n \geq 3$, a special n-triangle is a triangle with $n$ distinct numbers on each side such that the sum of the numbers on a side is the same for all sides. For instance, because $41 + 23 + 43 = 43 + 17 + 47 = 47 + 19 + 41$, the following is a special $3$-triangle: $$41$$$$23\text{ }\text{ }\text{ }\text{ }\text{ }19$$$$43\text{ }\text{ }\text{ }\text{ }\text{ }17\text{ }\text{ }\text{ }\text{ }\text{ }47$$ Note that a special $n$-triangle contains $3(n - 1)$ numbers. An infinite set $A$ of positive integers is a special set if, for each $n \geq 3$, the smallest $3(n - 1)$ numbers of $A$ can be used to form a special $n$-triangle. Show that the set of positive integers that are not multiples of $2023$ is a special set.

Let $A, B, C, D, E$ be five points on a circle such that $|AB| = |CD|$ and $|BC| = |DE|$. The segments $AD$ and $BE$ intersect at $F$. Let $M$ denote the midpoint of segment $CD$. Prove that the circle of center $M$ and radius $ME$ passes through the midpoint of segment $AF$.

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ with the property that $$f(x)f(y) = (xy - 1)^2f\left(\frac{x + y - 1}{xy - 1}\right)$$ for all real numbers $x, y$ with $xy \neq 1$.

The positive integers $a, b, c, d$ satisfy (i) $a + b + c + d = 2023$ (ii) $2023 \text{ } | \text{ } ab - cd$ (iii) $2023 \text{ } | \text{ } a^2 + b^2 + c^2 + d^2.$ Assuming that each of the numbers $a, b, c, d$ is divisible by $7$, prove that each of the numbers $a, b, c, d$ is divisible by $17$.

A positive integer is totally square is the sum of its digits (written in base $10$) is a square number. For example, $13$ is totally square because $1 + 3 = 2^2$, but $16$ is not totally square. Show that there are infinitely many positive integers that are not the sum of two totally square integers.

Aisling and Brendan take alternate moves in the following game. Before the game starts, the number $x = 2023$ is written on a piece of paper. Aisling makes the first move. A move from a positive integer $x$ consists of replacing $x$ either with $x + 1$ or with $x/p$ where $p$ is a prime factor of $x$. The winner is the first player to write $x = 1$. Determine whether Aisling or Brendan has a winning strategy for this game.

Suppose that $a, b, c$ are positive real numbers and $a + b + c = 3$. Prove that $$\frac{a+b}{c+2} + \frac{b+c}{a+2} + \frac{c+a}{b+2} \geq 2$$ and determine when equality holds.

The triangle $ABC$ has circumcentre $O$ and circumcircle $\Gamma$. Let $AI$ be a diameter of $\Gamma$. The ray $AI$ extends to intersect the circumcircle $\omega$ of $\triangle BOC$ for the second time at a point $P$. Let $AD$ and $IQ$ be perpendicular to $BC$, with $D$ and $Q$ on $BC$. Let $M$ be the midpoint of $BC$. (a) Prove that $|AD| \cdot |QI| = |CD| \cdot |CQ| = |BD| \cdot |BQ|$. (b) Prove that $IM$ is parallel to $PD$.

Caitlin and Donal play a game called Basketball Shoot-Out. The game consists of $10$ rounds. In each round, Caitlin and Donal both throw a ball simultaneously at each other's basket. If a player's ball falls into the basket, that player scores one point; otherwise, they score zero points. The scoreboard shows the complete sequence of points scored by each player in each of the $10$ rounds of the game. It turns out that Caitlin has scored at least as many points in total as Donal after every round of the game. Prove the number of possible scoreboards is divisible by $4$ but not by $8$.