Let $P \in \mathbb{R}[x]$ be a polynomial with real coefficients, and suppose $\deg(P) \ge 2$. For every $x \in \mathbb{R}$, let $\ell_x \subset \mathbb{R}^2$ denote the line tangent to the graph of $P$ at the point $(x, P(x))$.

(a) Suppose that the degree of $P$ is odd. Show that $\bigcup\limits_{x \in \mathbb{R}} \ell_x = \mathbb{R}^2$.

(b) Does there exist a polynomial of even degree for which the above equality still holds?

(proposed by Mike Daas, Max Planck Institute for Mathematics, Bonn)

Let $f : \mathbb{R} \to \mathbb{R}$ be a twice continuously differentiable function, and suppose that $\int_{-1}^{1} f(x)\,dx = 0$ and $f(1) = f(-1) = 1$. Prove that $$\int_{-1}^{1} (f''(x))^2\,dx \ge 15,$$ and find all such functions for which equality holds.

(proposed by Alberto Cagnetta, Università degli Studi di Udine, Italy)

Denote by $S$ the set of all real symmetric $2025 \times 2025$ matrices of rank 1 whose entries take values $-1$ or $+1$. Let $A, B \in S$ be matrices chosen independently uniformly at random. Find the probability that $A$ and $B$ commute, i.e. $AB = BA$.

(proposed by Marian Panţiruc, ”Gheorghe Asachi” Technical University of Iaşi, Romania)

Let $a$ be an even positive integer. Find all real numbers $x$ such that $$\tag{1} \left\lfloor \sqrt[a]{b^a + x} \cdot b^{a-1} \right\rfloor = b^a + \lfloor x/a \rfloor$$ holds for every positive integer $b$.

(Here $\lfloor x \rfloor$ denotes the largest integer that is no greater than $x$.)

(proposed by Yagub Aliyev, ADA University, Baku, Azerbaijan)

For a positive integer $n$, let $[n] = \{1, 2, \dots, n\}$. Denote by $S_n$ the set of all bijections from $[n]$ to $[n]$, and let $T_n$ be the set of all maps from $[n]$ to $[n]$. Define the order $\operatorname{ord}(\tau)$ of a map $\tau \in T_n$ as the number of distinct maps in the set $\{\tau, \tau \circ \tau, \tau \circ \tau \circ \tau, \dots\}$ where $\circ$ denotes composition. Finally, let $$f(n) = \max_{\tau \in S_n} \operatorname{ord}(\tau) \quad \text{and} \quad g(n) = \max_{\tau \in T_n} \operatorname{ord}(\tau).$$ Prove that $g(n) < f(n) + n^{0.501}$ for sufficiently large $n$.

(proposed by Fedor Petrov, St Petersburg State University)

Let $f : (0, \infty) \to \mathbb{R}$ be a continuously differentiable function, and let $b > a > 0$ be real numbers such that $f(a) = f(b) = k$. Prove that there exists a point $\xi \in (a, b)$ such that $$f(\xi) - \xi f'(\xi) = k.$$ (proposed by Alberto Cagnetta, Università degli Studi di Udine)

Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all nonempty subsets $M \subseteq \mathbb{Z}_{>0}$ satisfying both of the following properties:

(a) if $x \in M$, then $2x \in M$,

(b) if $x, y \in M$ and $x + y$ is even, then $\dfrac{x + y}{2} \in M$.

(proposed by Alexandr Bolbot, Novosibirsk State University)

For an $n \times n$ real matrix $A \in M_n(\mathbb{R})$, denote by $A^R$ its counter-clockwise $90^\circ$ rotation. For example, $$\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}^R = \begin{pmatrix} 3 & 6 & 9 \\ 2 & 5 & 8 \\ 1 & 4 & 7 \end{pmatrix}.$$ Prove that if $A = A^R$ then for any eigenvalue $\lambda$ of $A$, we have $\operatorname{Re} \lambda = 0$ or $\operatorname{Im} \lambda = 0$.

(proposed by Jan Kuś, University of Warwick)

Let $n$ be a positive integer. Consider the following random process which produces a sequence of $n$ distinct positive integers $X_1, X_2, \dots, X_n$.

First, $X_1$ is chosen randomly with $\mathsf{P}(X_1 = i) = 2^{-i}$ for every positive integer $i$. For $1 \le j \le n-1$, having chosen $X_1, \dots, X_j$, arrange the remaining positive integers in increasing order as $n_1 < n_2 < \cdots$, and choose $X_{j+1}$ randomly with $\mathsf{P}(X_{j+1} = n_i) = 2^{-i}$ for every positive integer $i$.

Let $Y_n = \max\{X_1, \dots, X_n\}$. Show that $$\mathsf{E}[Y_n] = \sum_{i=1}^{n} \frac{2^i}{2^i - 1}$$ where $\mathsf{E}[Y_n]$ is the expected value of $Y_n$.

(proposed by Jan Kuś and Jun Yan, University of Warwick)

For any positive integer $N$, let $S_N$ be the number of pairs of integers $1 \le a, b \le N$ such that the number $(a^2 + a)(b^2 + b)$ is a perfect square. Prove that the limit $$\lim_{N \to \infty} \frac{S_N}{N}$$ exists and find its value.

(proposed by Besfort Shala, University of Bristol)