Find the set of all possible values of the expression $\lfloor m^2+\sqrt{2} n \rfloor$, where $m$ and $n$ are positive integers. Note: The symbol $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.

For each positive integer $k$ we denote by $S(k)$ the sum of its digits, for example $S(132)=6$ and $S(1000)=1$. A positive integer $n$ is said to be $\textbf{fascinating}$ if it holds that $n = \frac{k}{S(k)}$ for some positive integer $k$. For example, the number $11$ is $\textbf{fascinating}$ since $11 = \frac{198}{S(198)} ($since $\frac{198}{S(198)}=\frac{198}{1+9+8}=\frac{198}{18} = 11)$. Prove that there exists a positive integer less than $2021$ and that it is not $\textbf{fascinating}$.

Let $ABC$ be a triangle and $D$ is a point in $BC$. The line $DA$ cuts the circumcircle of $ABC$ in the point $E$. Let $M$ and $N$ be the midpoints of $AB$ and $CD$, respectively. Let $F=MN\cap AD$ and $G\neq F$ is the point of intersection of the circumcircles of $\triangle DNF$ and $\triangle ECF$. Prove that $B,F$ and $G$ are collinears.

Let $n\ge 5$ be an integer. Consider $2n-1$ subsets $A_1, A_2, A_3, \ldots , A_{2n-1}$ of the set $\{ 1, 2, 3,\ldots , n \}$, these subsets have the property that each of them has $2$ elements (that is that is, for $1 \le i \le 2n-1$ it is true that $A_i$ has $2$ elements). Show that it is always possible to select $n$ of these subsets in such a way that the union of these $n$ subsets has at most $\frac{2}{3}n + 1$ elements in total.

Let $n\ge 2$ be an integer. They are given $n + 1$ red points in the plane. Prove that there exist $2n$ circles $C_1 , C_2 , \ldots , C_n , D_1 , D_2 , \ldots , D_n$ such that: $\bullet$ $C_1 , C_2 ,\ldots , C_n$ are concentric. $\bullet$ $D_1 , D_2 ,\ldots , D_n$ are concentric. $\bullet$ For $k = 1, 2, 3,\ldots , n$ the circles $C_k$ and $D_k$ are disjoint. $\bullet$ For $k = 1, 2, 3,\ldots , n$ it is true that $C_k$ contains exactly $k$ red dots in its interior and $D_k$ contains exactly $n + 1 - k$ red dots in its interior.

Prove that there are no positive integers $a_1, a_2, \ldots , a_{2021}$ (not necessarily distinct) such that for $k = 1, 2, 3, \ldots , 2021$ the number of elements in the set $$A_k = \{ j \in \mathbb{N} : 1 \le j \le 2021 \text{ and } a_j|k \}$$be exactly $a_k$.

Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$. Proposed by Dominik Burek and Tomasz Ciesla, Poland