$ABCD$ is a rectangle with side lengths $AB = CD = 1$ and $BC = DA = 2$. Let $ M$ be the midpoint of $AD$. Point $P$ lies on the opposite side of line $MB$ to $A$, such that triangle $MBP$ is equilateral. Find the value of $\angle PCB$.

Is it possible to pair up the numbers $0, 1, 2, 3,... , 61$ in such a way that when we sum each pair, the product of the $31$ numbers we get is a perfect f ifth power?

Find all real numbers$ x$ and $y$ such that $$x^2 + y^2 = 2$$$$\frac{x^2}{2 - y}+\frac{y^2}{2 - x}= 2.$$

On a table, there is an empty bag and a chessboard containing exactly one token on each square. Next to the table is a large pile that contains an unlimited supply of tokens. Using only the following types of moves what is the maximum possible number of tokens that can be in the bag? $\bullet$ Type 1: Choose a non-empty square on the chessboard that is not in the rightmost column. Take a token from this square and place it, along with one token from the pile, on the square immediately to its right. $\bullet$ Type 2: Choose a non-empty square on the chessboard that is not in the bottommost row. Take a token from this square and place it, along with one token from the pile, on the square immediately below it. $\bullet$ Type 3: Choose two adjacent non-empty squares. Remove a token from each and put them both into the bag.

A round-robin tournament is one where each team plays every other team exactly once. Five teams take part in such a tournament getting: $3$ points for a win, $1$ point for a draw and $0$ points for a loss. At the end of the tournament the teams are ranked from first to last according to the number of points. (a) Is it possible that at the end of the tournament, each team has a different number of points, and each team except for the team ranked last has exactly two more points than the next-ranked team? (b) Is this possible if there are six teams in the tournament instead?

Let a positive integer $n$ be given. Determine, in terms of $n$, the least positive integer $k$ such that among any $k$ positive integers, it is always possible to select a positive even number of them having sum divisible by $n$.

Let $M$ be the midpoint of side $BC$ of acute triangle $ABC$. The circle centered at $M$ passing through $A$ intersects the lines $AB$ and $AC$ again at $P$ and $Q$, respectively. The tangents to this circle at $P$ and $Q$ meet at $D$. Prove that the perpendicular bisector of $BC$ bisects segment $AD$.

Find all real numbers $x$ such that $-1 < x \le 2 $ and $$\sqrt{2 - x}+\sqrt{2 + 2x} =\sqrt{\frac{x^4 + 1}{x^2 + 1}}+ \frac{x + 3}{x + 1}.$$.

Find all integers $a, b$ such that $$a^2 + b = b^{2022}.$$

Find all triples $(a, b, c) $ of real numbers such that $a^2 + b^2 + c^2 = 1$ and $a(2b - 2a - c) \ge \frac12$.

Let $S$ be a set of $10$ positive integers. Prove that one can find two disjoint subsets $A =\{a_1, ..., a_k\}$ and $B = \{b_1, ... , b_k\}$ of $S$ with $|A| = |B|$ such that the sums $x =\frac{1}{a_1}+ ... +\frac{1}{a_k}$ and $y =\frac{1}{b_1}+ ... +\frac{1}{b_k}$ differ by less than $0.01$, i.e., $|x - y| < 1/100$.

Triangle $ABC$ is right-angled at $B$ and has incentre $I$. Points $D$, $E$ and $F$ are the points where the incircle of the triangle touches the sides $BC$, $AC$ and AB respectively. Lines $CI$ and $EF$ intersect at point $P$. Lines $DP$ and $AB$ intersect at point $Q$. Prove that $AQ = BF$.

The sequence $x_1, x_2, x_3, . . .$ is defined by $x_1 = 2022$ and $x_{n+1}= 7x_n + 5$ for all positive integers $n$. Determine the maximum positive integer $m$ such that $$\frac{x_n(x_n - 1)(x_n - 2) . . . (x_n - m + 1)}{m!}$$is never a multiple of $7$ for any positive integer $n$.