Let $A$ and $B$ be two sets of non-negative integers, define $A+B$ as the set of the values obtained when we sum any (one) element of the set $A$ with any (one) element of the set $B$. For instance, if $A=\{2,3\}$ and $B=\{0,1,2,5\}$ so $A+B=\{2,3,4,5,7,8\}$. Determine the least integer $k$ such that there is a pair of sets $A$ and $B$ of non-negative integers with $k$ and $2k$ elements, respectively, and $A+B=\{0,1,2,\dots, 2019,2020\}$

Find all the pairs $(a,b)$ of integer numbers such that: $\triangleright$ $a-b-1|a^2+b^2$ $\triangleright$ $\frac{a^2+b^2}{2ab-1}=\frac{20}{19}$

Let $ABC$ be a triangle with $AB<AC$ and $I$ be your incenter. Let $M$ and $N$ be the midpoints of the sides $BC$ and $AC$, respectively. If the lines $AI$ and $IN$ are perpendicular, prove that the line $AI$ is tangent to the circumcircle of $\triangle IMC$.

The function $f:\mathbb{N}\rightarrow \mathbb{N}$ is peruvian if it satifies the following two properties: $\triangleright f$ is strictly increasing. $\triangleright$ The numbers $a_1,a_2,a_3,\dots$ where $a_1=f(1)$ and $a_{n+1}=f(a_n)$ for every $n\geq 1$, are in arithmetic progression. Determine all peruvian functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that $f(1)=3$.

Let $AD$ be the diameter of a circle $\omega$ and $BC$ is a chord of $\omega$ which is perpendicular to $AD$. Let $M,N,P$ be points on the segments $AB,AC,BC$ respectively, such that $MP\parallel AC$ and $PN\parallel AB$. The line $MN$ cuts the line $PD$ in the point $Q$ and the angle bisector of $\angle MPN$ in the point $R$. Prove that the points $B,R,Q,C$ are concyclic.

A table $110\times 110$ is given, we define the distance between two cells $A$ and $B$ as the least quantity of moves to move a chess king from the cell $A$ to cell $B$. We marked $n$ cells on the table $110\times 110$ such that the distance between any two cells is not equal to $15$. Determine the greatest value of $n$.