Let $f : [a, b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$. Suppose that $f$ has infinitely many zeros, but there is no $x \in (a, b)$ with $f(x) = f'(x) = 0$.

(a) Prove that $f(a) f(b) = 0$.

(b) Give an example of such a function on $[0, 1]$.

(Proposed by Alexandr Bolbot, Novosibirsk State University)

Let $k$ and $n$ be positive integers. A sequence $(A_1, \dots, A_k)$ of $n \times n$ real matrices is preferred by Ivan the Confessor if $A_i^2 \ne 0$ for $1 \le i \le k$, but $A_i A_j = 0$ for $1 \le i, j \le k$ with $i \ne j$. Show that $k \le n$ in all preferred sequences, and give an example of a preferred sequence with $k = n$ for each $n$.

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

Let $n$ be a positive integer. Also let $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_n$ be real numbers such that $a_i + b_i > 0$ for $i = 1, 2, \dots, n$. Prove that $$\sum_{i=1}^{n} \frac{a_i b_i - b_i^2}{a_i + b_i} \le \frac{\sum\limits_{i=1}^{n} a_i \cdot \sum\limits_{i=1}^{n} b_i - \left( \sum\limits_{i=1}^{n} b_i \right)^2} {\sum\limits_{i=1}^{n} (a_i + b_i)}.$$ (Proposed by Daniel Strzelecki, Nicolaus Copernicus University in Toruń, Poland)

Let $n \ge k$ be positive integers, and let $\mathcal{F}$ be a family of finite sets with the following properties:

(i) $\mathcal{F}$ contains at least $\binom{n}{k} + 1$ distinct sets containing exactly $k$ elements;

(ii) for any two sets $A, B \in \mathcal{F}$, their union $A \cup B$ also belongs to $\mathcal{F}$.

Prove that $\mathcal{F}$ contains at least three sets with at least $n$ elements.

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

Let $S_n$ denote the set of permutations of the sequence $(1, 2, \dots, n)$. For every permutation $\pi = (\pi_1, \dots, \pi_n) \in S_n$, let $\mathrm{inv}(\pi)$ be the number of pairs $1 \le i < j \le n$ with $\pi_i > \pi_j$; i.e. the number of inversions in $\pi$. Denote by $f(n)$ the number of permutations $\pi \in S_n$ for which $\mathrm{inv}(\pi)$ is divisible by $n + 1$.

Prove that there exist infinitely many primes $p$ such that $f(p-1) > \frac{(p-1)!}{p}$, and infinitely many primes $p$ such that $f(p-1) < \frac{(p-1)!}{p}$.

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

Let $(x_1, x_2, \dots)$ be a sequence of positive real numbers satisfying $\displaystyle\sum_{n=1}^{\infty} \frac{x_n}{2n-1} = 1$. Prove that $$\sum_{k=1}^{\infty} \sum_{n=1}^{k} \frac{x_n}{k^2} \le 2.$$ (Proposed by Gerhard J. Woeginger, The Netherlands)

Today, Ivan the Confessor prefers continuous functions $f : [0, 1] \to \mathbb{R}$ satisfying $f(x) + f(y) \ge |x - y|$ for all pairs $x, y \in [0, 1]$. Find the minimum of $\int_0^1 f$ over all preferred functions.

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

Let $n$ be a positive integer, and denote by $\mathbb{Z}_n$ the ring of integers modulo $n$. Suppose that there exists a function $f : \mathbb{Z}_n \to \mathbb{Z}_n$ satisfying the following three properties:

(i) $f(x) \ne x$,

(ii) $f(f(x)) = x$,

(iii) $f(f(f(x + 1) + 1) + 1) = x$ for all $x \in \mathbb{Z}_n$.

Prove that $n \equiv 2 \pmod 4$.

(Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Germany)

Let $k$ be a positive integer. For each nonnegative integer $n$, let $f(n)$ be the number of solutions $(x_1, \dots, x_k) \in \mathbb{Z}^k$ of the inequality $|x_1| + \dots + |x_k| \le n$. Prove that for every $n \ge 1$, we have $f(n-1) f(n+1) \le f(n)^2$.

(Proposed by Esteban Arreaga, Renan Finder and José Madrid, IMPA, Rio de Janeiro)

Let $A$ be a $n \times n$ complex matrix whose eigenvalues have absolute value at most 1. Prove that $$\|A^n\| \le \frac{n}{\ln 2} \|A\|^{n-1}.$$ (Here $\|B\| = \sup\limits_{\|x\| \le 1} \|Bx\|$ for every $n \times n$ matrix $B$ and $\|x\| = \sqrt{\sum\limits_{i=1}^{n} |x_i|^2}$ for every complex vector $x \in \mathbb{C}^n$.)

(Proposed by Ian Morris and Fedor Petrov, St. Petersburg State University)