In each cell of a chessboard with $2$ rows and $2019$ columns a real number is written so that: There are no two numbers written in the first row that are equal to each other. The numbers written in the second row coincide with (in some another order) the numbers written in the first row. The two numbers written in each column are different and they add up to a rational number. Determine the maximum quantity of irrational numbers that can be in the chessboard.

A power is a positive integer of the form $a^k$, where $a$ and $k$ are positive integers with $k\geq 2$. Let $S$ be the set of positive integers which cannot be expressed as sum of two powers (for example, $4,\ 7,\ 15$ and $27$ are elements of $S$). Determine whether the set $S$ has a finite or infinite number of elements.

Let $I,\ O$ and $\Gamma$ be the incenter, circumcenter and the circumcircle of triangle $ABC$, respectively. Line $AI$ meets $\Gamma$ at $M$ $(M\neq A)$. The circumference $\omega$ is tangent internally to $\Gamma$ at $T$, and is tangent to the lines $AB$ and $AC$. The tangents through $A$ and $T$ to $\Gamma$ intersect at $P$. Lines $PI$ and $TM$ meet at $Q$. Prove that the lines $QA$ and $MO$ meet at a point on $\Gamma$.

Let $k\geq 0$ an integer. The sequence $a_0,\ a_1,\ a_2, \ a_3, \ldots$ is defined as follows: $a_0=k$ For $n\geq 1$, we have that $a_n$ is the smallest integer greater than $a_{n-1}$ so that $a_n+a_{n-1}$ is a perfect square. Prove that there are exactly $\left \lfloor{\sqrt{2k}} \right \rfloor$ positive integers that cannot be written as the difference of two elements of such a sequence. Note. If $x$ is a real number, $\left \lfloor{x} \right \rfloor$ denotes the greatest integer smaller or equal than $x$.

Let $m$ and $n$ two given integers. Ana thinks of a pair of real numbers $x$, $y$ and then she tells Beto the values of $x^m+y^m$ and $x^n+y^n$, in this order. Beto's goal is to determine the value of $xy$ using that information. Find all values of $m$ and $n$ for which it is possible for Beto to fulfill his wish, whatever numbers that Ana had chosen.

Let $p$ and $q$ two positive integers. Determine the greatest value of $n$ for which there exists sets $A_1,\ A_2,\ldots,\ A_n$ and $B_1,\ B_2,\ldots,\ B_n$ such that: The sets $A_1,\ A_2,\ldots,\ A_n$ have $p$ elements each one. The sets $B_1,\ B_2,\ldots,\ B_n$ have $q$ elements each one. For all $1\leq i,\ j \leq n$, sets $A_i$ and $B_j$ are disjoint if and only if $i=j$.