Middle European Mathematical Olympiad 2021

Determine all real numbers $A$ such that every sequence of non-zero real numbers $x_1, x_2, \ldots$ satisfying

$$x_{n+1} = A - \frac{1}{x_n}$$

for every integer $n \geqslant 1$, has only finitely many negative terms.

Let $m$ and $n$ be positive integers. Some squares of an $m \times n$ board are coloured red. A sequence $a_1, a_2, \ldots, a_{2r}$ of $2r \geqslant 4$ pairwise distinct red squares is called a bishop circuit if for every $k \in \{1, \ldots, 2r\}$, the squares $a_k$ and $a_{k+1}$ lie on a diagonal, but the squares $a_k$ and $a_{k+2}$ do not lie on a diagonal (here $a_{2r+1} = a_1$ and $a_{2r+2} = a_2$).

In terms of $m$ and $n$, determine the maximum possible number of red squares on an $m \times n$ board without a bishop circuit.

(Remark. Two squares lie on a diagonal if the line passing through their centres intersects the sides of the board at an angle of $45°$.)

Let $ABC$ be an acute triangle and $D$ an interior point of segment $BC$. Points $E$ and $F$ lie in the half-plane determined by the line $BC$ containing $A$ such that $DE$ is perpendicular to $BE$ and $DE$ is tangent to the circumcircle of $ACD$, while $DF$ is perpendicular to $CF$ and $DF$ is tangent to the circumcircle of $ABD$. Prove that the points $A$, $D$, $E$ and $F$ are concyclic.

Let $n \geqslant 3$ be an integer. Zagi the squirrel sits at a vertex of a regular $n$-gon. Zagi plans to make a journey of $n - 1$ jumps such that in the $i$-th jump, it jumps by $i$ edges clockwise, for $i \in \{1, \ldots, n - 1\}$. Prove that if after $\lceil \frac{n}{2} \rceil$ jumps Zagi has visited $\lceil \frac{n}{2} \rceil + 1$ distinct vertices, then after $n - 1$ jumps Zagi will have visited all of the vertices.

(Remark. For a real number $x$, we denote by $\lceil x \rceil$ the smallest integer larger or equal to $x$.)

Determine all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that the inequality

$$f(x^2) - f(y^2) \leqslant (f(x) + y)(x - f(y))$$

holds for all real numbers $x$ and $y$.

Given a positive integer $n$, we say that a polynomial $P$ with real coefficients is $n$-pretty if the equation $P(\lfloor x \rfloor) = \lfloor P(x) \rfloor$ has exactly $n$ real solutions. Show that for each positive integer $n$

(a) there exists an $n$-pretty polynomial;

(b) any $n$-pretty polynomial has a degree of at least $\frac{2n + 1}{3}$.

(Remark. For a real number $x$, we denote by $\lfloor x \rfloor$ the largest integer smaller than or equal to $x$.)

Let $n$, $b$ and $c$ be positive integers. A group of $n$ pirates wants to fairly split their treasure. The treasure consists of $c \cdot n$ identical coins distributed over $b \cdot n$ bags, of which at least $n - 1$ bags are initially empty. Captain Jack inspects the contents of each bag and then performs a sequence of moves. In one move, he can take any number of coins from a single bag and put them into one empty bag. Prove that no matter how the coins are initially distributed, Jack can perform at most $n - 1$ moves and then split the bags among the pirates such that each pirate gets $b$ bags and $c$ coins.

Let $n$ be a positive integer. Prove that in a regular $6n$-gon, we can draw $3n$ diagonals with pairwise distinct ends and partition the drawn diagonals into $n$ triplets so that:

• the diagonals in each triplet intersect in one interior point of the polygon and
• all these $n$ intersection points are distinct.

Let $AD$ be the diameter of the circumcircle of an acute triangle $ABC$. The lines through $D$ parallel to $AB$ and $AC$ meet lines $AC$ and $AB$ in points $E$ and $F$, respectively. Lines $EF$ and $BC$ meet at $G$. Prove that $AD$ and $DG$ are perpendicular.

Let $ABC$ be a triangle and let $M$ be the midpoint of the segment $BC$. Let $X$ be a point on the ray $AB$ such that $2\angle CXA = \angle CMA$. Let $Y$ be a point on the ray $AC$ such that $2\angle AYB = \angle AMB$. The line $BC$ intersects the circumcircle of the triangle $AXY$ at $P$ and $Q$, such that the points $P$, $B$, $C$, and $Q$ lie in this order on the line $BC$. Prove that $PB = QC$.

Find all pairs $(n,p)$ of positive integers such that $p$ is prime and

$$1 + 2 + \cdots + n = 3 \cdot (1^2 + 2^2 + \cdots + p^2).$$

Prove that there are infinitely many positive integers $n$ such that $n^2$ written in base $4$ contains only digits $1$ and $2$.