1.
Let $f\colon [a,b]\to\mathbb{R}$ 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

2.
Let $k$ and $n$ be positive integers. A sequence $(A_1,\ldots,A_k)$
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 $A_iA_j=0$ 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

3.
Let $n$ be a positive integer. Also let $a_1,a_2,\ldots,a_n$ and
$b_1,b_2,\ldots,b_n$ be real numbers such that $a_i+b_i>0$ for
$i=1,2,\ldots,n$. Prove that
$$
\sum_{i=1}^n \frac{a_ib_i-b_i^2}{a_i+b_i}\leq \frac{\sum\limits_{i=1}^n a_i \cdot \sum\limits_{i=1}^n b_i -\left(\sum\limits_{i=1}^n b_i\right)^2}{\sum\limits_{i=1}^n (a_i+b_i)}.
$$

Proposed by Daniel Strzelecki, Nicolaus Copernicus
University in TorĂșn, Poland

4.
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

5.
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 $1\le i\lt j\le n$
with $\pi_i\gt \pi_j$; 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 $f(p-1)\gt \dfrac{(p-1)!}p$, and
infinitely many primes $p$ such that $f(p-1)\lt \dfrac{(p-1)!}p$.

Proposed by Fedor Petrov, St. Petersburg State
University