Determine all complex numbers $\lambda$ for which there exist a positive integer $n$ and a real $n \times n$ matrix $A$ such that $A^2 = A^T$ and $\lambda$ is an eigenvalue of $A$.

(Proposed by Alexandr Bolbot, Novosibirsk State University)

Let $f : \mathbb{R} \to (0, \infty)$ be a differentiable function, and suppose that there exists a constant $L > 0$ such that $$\bigl| f'(x) - f'(y) \bigr| \le L \bigl| x - y \bigr|$$ for all $x, y$. Prove that $$\bigl( f'(x) \bigr)^2 < 2 L f(x)$$ holds for all $x$.

(Proposed by Jan Šustek, University of Ostrava)

For any positive integer $m$, denote by $P(m)$ the product of positive divisors of $m$ (e.g. $P(6) = 36$). For every positive integer $n$ define the sequence $$a_1(n) = n, \qquad a_{k+1}(n) = P(a_k(n)) \quad (k = 1, 2, \dots, 2016).$$ Determine whether for every set $S \subseteq \{1, 2, \dots, 2017\}$, there exists a positive integer $n$ such that the following condition is satisfied:

For every $k$ with $1 \le k \le 2017$, the number $a_k(n)$ is a perfect square if and only if $k \in S$.

(Proposed by Matko Ljulj, University of Zagreb)

There are $n$ people in a city, and each of them has exactly 1000 friends (friendship is always symmetric). Prove that it is possible to select a group $S$ of people such that at least $n/2017$ persons in $S$ have exactly two friends in $S$.

(Proposed by Rooholah Majdodin and Fedor Petrov, St. Petersburg State University)

Let $k$ and $n$ be positive integers with $n \ge k^2 - 3k + 4$, and let $$f(z) = z^{n-1} + c_{n-2} z^{n-2} + \dots + c_0$$ be a polynomial with complex coefficients such that $$c_0 c_{n-2} = c_1 c_{n-3} = \dots = c_{n-2} c_0 = 0.$$ Prove that $f(z)$ and $z^n - 1$ have at most $n - k$ common roots.

(Proposed by Vsevolod Lev and Fedor Petrov, St. Petersburg State University)

Let $f : [0; +\infty) \to \mathbb{R}$ be a continuous function such that $\lim\limits_{x \to +\infty} f(x) = L$ exists (it may be finite or infinite). Prove that $$\lim_{n \to \infty} \int_0^1 f(nx)\,dx = L.$$ (Proposed by Alexandr Bolbot, Novosibirsk State University)

Let $p(x)$ be a nonconstant polynomial with real coefficients. For every positive integer $n$, let $$q_n(x) = (x+1)^n p(x) + x^n p(x+1).$$ Prove that there are only finitely many numbers $n$ such that all roots of $q_n(x)$ are real.

(Proposed by Alexandr Bolbot, Novosibirsk State University)

Define the sequence $A_1, A_2, \dots$ of matrices by the following recurrence: $$A_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad A_{n+1} = \begin{pmatrix} A_n & I_{2^n} \\ I_{2^n} & A_n \end{pmatrix} \quad (n = 1, 2, \dots)$$ where $I_m$ is the $m \times m$ identity matrix.

Prove that $A_n$ has $n + 1$ distinct integer eigenvalues $\lambda_0 < \lambda_1 < \dots < \lambda_n$ with multiplicities $\binom{n}{0}, \binom{n}{1}, \dots, \binom{n}{n}$, respectively.

(Proposed by Snježana Majstorović, University of J. J. Strossmayer in Osijek, Croatia)

Define the sequence $f_1, f_2, \dots : [0, 1) \to \mathbb{R}$ of continuously differentiable functions by the following recurrence: $$f_1 = 1; \qquad f'_{n+1} = f_n f_{n+1} \text{ on } (0, 1), \text{ and } f_{n+1}(0) = 1.$$ Show that $\lim\limits_{n \to \infty} f_n(x)$ exists for every $x \in [0, 1)$ and determine the limit function.

(Proposed by Tomáš Bárta, Charles University, Prague)

Let $K$ be an equilateral triangle in the plane. Prove that for every $p > 0$ there exists an $\varepsilon > 0$ with the following property: If $n$ is a positive integer, and $T_1, \dots, T_n$ are non-overlapping triangles inside $K$ such that each of them is homothetic to $K$ with a negative ratio, and $$\sum_{\ell=1}^{n} \operatorname{area}(T_\ell) > \operatorname{area}(K) - \varepsilon,$$ then $$\sum_{\ell=1}^{n} \operatorname{perimeter}(T_\ell) > p.$$ (Proposed by Fedor Malyshev, Steklov Math. Inst. and Ilya Bogdanov, MIPT, Moscow)