Determine all pairs $(a, b)$ of real numbers for which there exists a unique symmetric $2 \times 2$ matrix $M$ with real entries satisfying $\operatorname{trace}(M) = a$ and $\det(M) = b$.

(Proposed by Stephan Wagner, Stellenbosch University)

Consider the following sequence $$(a_n)_{n=1}^{\infty} = (1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, \dots).$$ Find all pairs $(\alpha, \beta)$ of positive real numbers such that $\displaystyle\lim_{n \to \infty} \frac{\sum\limits_{k=1}^{n} a_k}{n^\alpha} = \beta$.

(Proposed by Tomas Barta, Charles University, Prague)

Let $n$ be a positive integer. Show that there are positive real numbers $a_0, a_1, \dots, a_n$ such that for each choice of signs the polynomial $$\pm a_n x^n \pm a_{n-1} x^{n-1} \pm \dots \pm a_1 x \pm a_0$$ has $n$ distinct real roots.

(Proposed by Stephan Neupert, TUM, München)

Let $n > 6$ be a perfect number, and let $n = p_1^{e_1} \cdots p_k^{e_k}$ be its prime factorisation with $1 < p_1 < \dots < p_k$. Prove that $e_1$ is an even number.

A number $n$ is perfect if $s(n) = 2n$, where $s(n)$ is the sum of the divisors of $n$.

(Proposed by Javier Rodrigo, Universidad Pontificia Comillas)

Let $A_1 A_2 \dots A_{3n}$ be a closed broken line consisting of $3n$ line segments in the Euclidean plane. Suppose that no three of its vertices are collinear, and for each index $i = 1, 2, \dots, 3n$, the triangle $A_i A_{i+1} A_{i+2}$ has counterclockwise orientation and $\angle A_i A_{i+1} A_{i+2} = 60^\circ$, using the notation $A_{3n+1} = A_1$ and $A_{3n+2} = A_2$. Prove that the number of self-intersections of the broken line is at most $\frac{3}{2} n^2 - 2n + 1$.

(Proposed by Martin Langer)

For a positive integer $x$, denote its $n^{\text{th}}$ decimal digit by $d_n(x)$, i.e. $d_n(x) \in \{0, 1, \dots, 9\}$ and $x = \sum_{n=1}^{\infty} d_n(x) 10^{n-1}$. Suppose that for some sequence $\bigl( a_n \bigr)_{n=1}^{\infty}$, there are only finitely many zeros in the sequence $\bigl( d_n(a_n) \bigr)_{n=1}^{\infty}$. Prove that there are infinitely many positive integers that do not occur in the sequence $\bigl( a_n \bigr)_{n=1}^{\infty}$.

(Proposed by Alexander Bolbot, State University, Novosibirsk)

Let $A = (a_{ij})_{i,j=1}^{n}$ be a symmetric $n \times n$ matrix with real entries, and let $\lambda_1, \lambda_2, \dots, \lambda_n$ denote its eigenvalues. Show that $$\sum_{1 \le i < j \le n} a_{ii} a_{jj} \ge \sum_{1 \le i < j \le n} \lambda_i \lambda_j,$$ and determine all matrices for which equality holds.

(Proposed by Martin Niepel, Comenius University, Bratislava)

Let $f(x) = \frac{\sin x}{x}$, for $x > 0$, and let $n$ be a positive integer. Prove that $\left| f^{(n)}(x) \right| < \frac{1}{n+1}$, where $f^{(n)}$ denotes the $n^{\text{th}}$ derivative of $f$.

(Proposed by Alexander Bolbot, State University, Novosibirsk)

We say that a subset of $\mathbb{R}^n$ is $k$-almost contained by a hyperplane if there are less than $k$ points in that set which do not belong to the hyperplane. We call a finite set of points $k$-generic if there is no hyperplane that $k$-almost contains the set. For each pair of positive integers $k$ and $n$, find the minimal number $d(k, n)$ such that every finite $k$-generic set in $\mathbb{R}^n$ contains a $k$-generic subset with at most $d(k, n)$ elements.

(Proposed by Shachar Carmeli, Weizmann Inst. and Lev Radzivilovsky, Tel Aviv Univ.)

For every positive integer $n$, denote by $D_n$ the number of permutations $(x_1, \dots, x_n)$ of $(1, 2, \dots, n)$ such that $x_j \ne j$ for every $1 \le j \le n$. For $1 \le k \le \frac{n}{2}$, denote by $\Delta(n, k)$ the number of permutations of $(1, 2, \dots, n)$ such that $x_i = k + i$ for every $1 \le i \le k$ and $x_j \ne j$ for every $1 \le j \le n$. Prove that $$\Delta(n, k) = \sum_{i=0}^{k-1} \binom{k-1}{i} \frac{D_{(n+1)-(k+i)}}{n - (k+i)}.$$ (Proposed by Combinatorics; Ferdowsi University of Mashhad, Iran; Mirzavaziri)