Problem 1. Let $x_1, x_2, x_3, \ldots$ be a sequence of positive integers defined as follows: $x_1=1$ and for each $n \geqslant 1$ we have
$$
x_{n+1}=x_n+\left\lfloor\sqrt{x_n}\right\rfloor
$$
Determine all positive integers $m$ for which $x_n=m^2$ for some $n \geqslant 1$. (Here $\lfloor x\rfloor$ denotes the greatest integer less or equal to $x$ for every real number $x$.)
Problem 2. Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying the following condition: for each $k>2026$, the number $f(k)$ equals the maximum number of times a number appears in the list $f(1), f(2), \ldots, f(k-1)$. Prove that $f(n)=f(n+f(n))$ for infinitely many $n \in \mathbb{N}$.
(Here $\mathbb{N}$ denotes the set ${1,2,3, \ldots}$ of positive integers.)
Problem 3. Let $A B C$ be an acute-angled scalene triangle with circumcircle $\Gamma$. Let $M$ be the midpoint of $B C$ and $N$ be the midpoint of the minor arc $\overparen{B C}$ of $\Gamma$. Points $P$ and $Q$ lie on segments $A B$ and $A C$ respectively such that $B P=B N$ and $C Q=C N$. Point $K \neq N$ lies on line $A N$ with $M K=M N$. Prove that $\angle P K Q=90^{\circ}$.
Problem 4. Two integers $a$ and $b$ are called companions if every prime number $p$ either divides both or none of $a, b$. Determine all functions $f: \mathbb{N}_0 \rightarrow \mathbb{N}_0$ such that $f(0)=0$ and the numbers $f(m)+n$ and $f(n)+m$ are companions for all $m, n \in \mathbb{N}_0$.
(Here $\mathbb{N}_0$ denotes the set of all non-negative integers.)
Problem 5. Three lines $\ell_1, \ell_2, \ell_3$ form an acute angled triangle $\mathcal{T}$ in the plane. Point $P$ lies in the interior of $\mathcal{T}$. Let $\tau_i$ denote the transformation of the plane such that the image $\tau_i(X)$ of any point $X$ in the plane is the reflection of $X$ in $\ell_i$, for each $i \in{1,2,3}$. Denote by $P_{i j k}$ the point $\tau_k\left(\tau_j\left(\tau_i(P)\right)\right)$ for each permutation $(i, j, k)$ of $(1,2,3)$.
Prove that $P_{123}, P_{132}, P_{213}, P_{231}, P_{312}, P_{321}$ are concyclic if and only if $P$ coincides with the orthocentre of $\mathcal{T}$.
Problem 6. Two decks $\mathcal{A}$ and $\mathcal{B}$ of 40 cards each are placed on a table at noon. Every minute thereafter, we pick the top cards $a \in \mathcal{A}$ and $b \in \mathcal{B}$ and perform a duel.
For any two cards $a \in \mathcal{A}$ and $b \in \mathcal{B}$, each time $a$ and $b$ duel, the outcome remains the same and is independent of all other duels. A duel has three possible outcomes:
The process ends when both decks are empty. A process is called a game if it ends. Prove that the maximum time a game can last equals 356 hours.
