Find all functions $f : \mathbb{R} \to \mathbb{R}$ that have a continuous second derivative and for which the equality $f(7x + 1) = 49 f(x)$ holds for all $x \in \mathbb{R}$.

(proposed by Alex Avdiushenko, Neapolis University Paphos, Cyprus)

Let $A$, $B$ and $C$ be $n \times n$ matrices with complex entries satisfying $$A^2 = B^2 = C^2 \quad \text{and} \quad B^3 = ABC + 2I.$$ Prove that $A^6 = I$.

(proposed by Mike Daas, Universiteit Leiden)

Find all polynomials $P$ in two variables with real coefficients satisfying the identity $$P(x, y) P(z, t) = P(xz - yt, xt + yz).$$ (proposed by Giorgi Arabidze, Free University of Tbilisi, Georgia)

Let $p$ be a prime number and let $k$ be a positive integer. Suppose that the numbers $a_i = i^k + i$ for $i = 0, 1, \dots, p-1$ form a complete residue system modulo $p$. What is the set of possible remainders of $a_2$ upon division by $p$?

(proposed by Tigran Hakobyan, Yerevan State University, Armenia)

Fix positive integers $n$ and $k$ such that $2 \le k \le n$ and a set $M$ consisting of $n$ fruits. A permutation is a sequence $x = (x_1, x_2, \dots, x_n)$ such that $\{x_1, \dots, x_n\} = M$. Ivan prefers some (at least one) of these permutations. He realized that for every preferred permutation $x$, there exist $k$ indices $i_1 < i_2 < \dots < i_k$ with the following property: for every $1 \le j < k$, if he swaps $x_{i_j}$ and $x_{i_{j+1}}$, he obtains another preferred permutation.

Prove that he prefers at least $k!$ permutations.

(proposed by Ivan Mitrofanov, École Normale Superieur Paris)

Ivan writes the matrix $\begin{pmatrix} 2 & 3 \\ 2 & 4 \end{pmatrix}$ on the board. Then he performs the following operation on the matrix several times:

• he chooses a row or a column of the matrix, and
• he multiplies or divides the chosen row or column entry-wise by the other row or column, respectively.

(proposed by Alex Avdiushenko, Neapolis University Paphos, Cyprus)

Let $V$ be the set of all continuous functions $f : [0, 1] \to \mathbb{R}$, differentiable on $(0, 1)$, with the property that $f(0) = 0$ and $f(1) = 1$. Determine all $\alpha \in \mathbb{R}$ such that for every $f \in V$, there exists some $\xi \in (0, 1)$ such that $$f(\xi) + \alpha = f'(\xi).$$ (proposed by Mike Daas, Leiden University)

Let $T$ be a tree with $n$ vertices; that is, a connected simple graph on $n$ vertices that contains no cycle. For every pair $u, v$ of vertices, let $d(u, v)$ denote the distance between $u$ and $v$, that is, the number of edges in the shortest path in $T$ that connects $u$ with $v$.

Consider the sums $$W(T) = \sum_{\substack{\{u,v\} \subseteq V(T) \\ u \ne v}} d(u, v) \quad \text{and} \quad H(T) = \sum_{\substack{\{u,v\} \subseteq V(T) \\ u \ne v}} \frac{1}{d(u, v)}.$$ Prove that $$W(T) \cdot H(T) \ge \frac{(n-1)^3 (n+2)}{4}.$$ (proposed by Slobodan Filipovski, University of Primorska, Koper)

We say that a real number $V$ is good if there exist two closed convex subsets $X$, $Y$ of the unit cube in $\mathbb{R}^3$, with volume $V$ each, such that for each of the three coordinate planes (that is, the planes spanned by any two of the three coordinate axes), the projections of $X$ and $Y$ onto that plane are disjoint.

Find $\sup \{ V \mid V \text{ is good} \}$.

(proposed by Josef Tkadlec and Arseniy Akopyan)

For every positive integer $n$, let $f(n)$, $g(n)$ be the minimal positive integers such that $$1 + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} = \frac{f(n)}{g(n)}.$$ Determine whether there exists a positive integer $n$ for which $g(n) > n^{0.999 n}$.

(proposed by Fedor Petrov, St. Petersburg State University)