Show that the equation $\cos (\cos x) = \sin (\sin x)$ has no real solutions.

(proposed by Alexander Slávik, Charles University, Prague)

Let $n$ be a positive integer. Suppose that $A$ and $B$ are $n \times n$ matrices with real entries such that $$A^{\top} A + B B^{\top} = A B + B A,$$ where $X^{\top}$ denotes the transpose of matrix $X$.

Does this imply that $A B = B A$?

(proposed by Nikolaos Kolliopoulos, University of Cyprus)

Consider a deck of $n \geq 2$ cards labeled $1, 2, \ldots, n$. An alternating shuffle of the deck is performed as follows. We split the deck into two non-empty stacks. We then sort the first stack in increasing order, and the second stack in decreasing order. Finally, we alternately take cards from the first and second stacks (starting with the first). If one of the stacks runs out, the remaining cards from the other stack are placed at the end. How many different final orders of the deck can be obtained in this way?

Example: Suppose $n = 6$, the first stack is $A = (1, 3)$, and the second stack is $B = (6, 5, 4, 2)$. Then the order resulting from the alternating shuffle is $(1, 6, 3, 5, 4, 2)$.

(proposed by Daniel Volostnov, Neapolis University Paphos, Cyprus)

Let $x_{1} > 0$. Define the sequence $\left\{ x_{n} \right\}$ by the recurrence $$x_{n+1} = \arctan \left( \frac{x_{1} + x_{2} + \cdots + x_{n}}{n} \right) \text{ for all } n \geq 1.$$ Find $\lim\limits_{n \rightarrow \infty} x_{n} \sqrt{\ln n}$, where $\ln x$ denotes the natural logarithm of $x$.

(proposed by Wanlong Han, Henan, China)

Prove that there exists a constant $C > 0$ such that for every pair $A, B$ of positive integers, there is a real polynomial $p(x)$ with $$p(0)^{2} > \sum_{i=1}^{A} p(-i)^{2} + \sum_{i=1}^{B} p(i)^{2} \quad \text{and} \quad \operatorname{deg} p < C \sqrt{A B}.$$

(proposed by Géza Kós, Loránd Eötvös University, Budapest)

• (a) Is there a differentiable function $f \colon \mathbb{R} \rightarrow \mathbb{R}$ such that $$f^{\prime}(f(x)) = x$$ for every $x \in \mathbb{R}$?
• (b) Is there a differentiable function $f \colon \mathbb{R} \rightarrow \mathbb{R}$ such that $$f^{\prime}(f(x)) = |x|$$ for every $x \in \mathbb{R}$?

(proposed by Nikolaos Kolliopoulos, University of Cyprus)

For a continuous function $f \colon [0,1] \rightarrow \mathbb{R}$, let $S(f)$ be the union of all straight segments in the plane joining points $(x, 0)$ and $(f(x), 1)$, where $x \in [0,1]$. Let $A(f)$ be the area of $S(f)$. Find the infimum of $A(f)$ over all continuous $f$.

(proposed by David Preiss, University of Warwick, UK)

Let $n \geq 5$, and suppose that $A = \left( a_{i j} \right)$ is a real symmetric $n \times n$ matrix such that $$a_{i i} = 0 \quad \text{and} \quad a_{i j} \in \{-1, 1\} \text{ for } i \neq j.$$ Assume that the scalar products of any two distinct rows of $A$ have the same value. Let $\lambda_{1}, \ldots, \lambda_{n}$ be the eigenvalues of $A$. Prove that $$\sum_{i=1}^{n} \left| \lambda_{i} \right| \geq 2 n - 2$$ and determine all matrices for which equality holds.

(proposed by Slobodan Filipovski, University of Primorska, Koper)

Let $a_{1}, a_{2}, \ldots$ be an infinite sequence of positive real numbers satisfying $$a_{1} + a_{2} + \cdots + a_{2n-1} = a_{n}^{2}$$ for all positive integers $n$. Prove that $a_{n} \geq 2 n - 1$ for all positive integers $n$.

(proposed by Ilya I. Bogdanov, MIPT, Moscow and Aleksandr Kuznetsov, SPbU, Saint Petersburg)

An infinite chessboard of size $d > 0$ is obtained by colouring the interiors of the squares of an infinite square grid of side length $d$ alternately white and black following the usual chessboard pattern. The points belonging to the grid lines have neither colour and the grid may be translated and rotated arbitrarily in the plane.

Is it true that for any finite set of points $p_{1}, \ldots, p_{n}$ in the plane, there exist $d \in (0,1)$ and an infinite chessboard of size $d$ such that all the points $p_{i}$ lie in white squares?

(proposed by David Hruška, Czech Academy of Sciences, Prague)