Let $f : [0, 1] \to (0, \infty)$ be an integrable function such that $f(x) \cdot f(1 - x) = 1$ for all $x \in [0, 1]$. Prove that $$\int_0^1 f(x)\,dx \ge 1.$$ (proposed by Mike Daas, Universiteit Leiden)

Let $n$ be a positive integer. Find all $n \times n$ real matrices $A$ with only real eigenvalues satisfying $$A + A^k = A^T$$ for some integer $k \ge n$.

($A^T$ denotes the transpose of $A$.)

(proposed by Camille Mau, Nanyang Technological University)

Let $p$ be a prime number. A flea is staying at point 0 of the real line. At each minute, the flea has three possibilities: to stay at its position, or to move by 1 to the left or to the right. After $p - 1$ minutes, it wants to be at 0 again. Denote by $f(p)$ the number of its strategies to do this (for example, $f(3) = 3$: it may either stay at 0 for the entire time, or go to the left and then to the right, or go to the right and then to the left). Find $f(p)$ modulo $p$.

(proposed by Fedor Petrov, St. Petersburg)

Let $n > 3$ be an integer. Let $\Omega$ be the set of all triples of distinct elements of $\{1, 2, \dots, n\}$. Let $m$ denote the minimal number of colours which suffice to colour $\Omega$ so that whenever $1 \le a < b < c < d \le n$, the triples $\{a, b, c\}$ and $\{b, c, d\}$ have different colours. Prove that $$\frac{1}{100} \log\log n \leqslant m \leqslant 100 \log\log n.$$ (proposed by Danila Cherkashin, St. Petersburg)

We colour all the sides and diagonals of a regular polygon $P$ with 43 vertices either red or blue in such a way that every vertex is an endpoint of 20 red segments and 22 blue segments. A triangle formed by vertices of $P$ is called monochromatic if all of its sides have the same colour. Suppose that there are 2022 blue monochromatic triangles. How many red monochromatic triangles are there?

(proposed by Mike Daas, Universiteit Leiden)

Let $p > 2$ be a prime number. Prove that there is a permutation $(x_1, x_2, \dots, x_{p-1})$ of the numbers $(1, 2, \dots, p-1)$ such that $$x_1 x_2 + x_2 x_3 + \dots + x_{p-2} x_{p-1} \equiv 2 \pmod p.$$ (proposed by Giorgi Arabidze, Tbilisi Free University, Georgia)

Let $A_1, A_2, \dots, A_k$ be $n \times n$ idempotent complex matrices such that $$A_i A_j = -A_j A_i \quad \text{for all } i \ne j.$$ Prove that at least one of the given matrices has rank $\le \dfrac{n}{k}$.

(A matrix $A$ is called idempotent if $A^2 = A$.)

(proposed by Danila Belousov, Novosibirsk)

Let $n, k \ge 3$ be integers, and let $S$ be a circle. Let $n$ blue points and $k$ red points be chosen uniformly and independently at random on the circle $S$. Denote by $F$ the intersection of the convex hull of the red points and the convex hull of the blue points. Let $m$ be the number of vertices of the convex polygon $F$ (in particular, $m = 0$ when $F$ is empty). Find the expected value of $m$.

(proposed by Fedor Petrov, St. Petersburg)