A single player game has the following rules: initially, there are $10$ piles of stones with $1,2,...,10$ stones, respectively. A movement consists on making one of the following operations: i) to choose $2$ piles, both of them with at least $2$ stones, combine them and then add $2$ stones to the new pile; ii) to choose a pile with at least $4$ stones, remove $2$ stones from it, and then split it into two piles with amount of piles to be chosen by the player. The game continues until is not possible to make an operation. Show that the number of piles with one stone in the end of the game is always the same, no matter how the movements are made.

Let $ABC$ be an acute triangle, with $AB<AC$. Let $K$ be the midpoint of the arch $BC$ that does not contain $A$ and let $P$ be the midpoint of $BC$. Let $I_B,I_C$ be the $B$-excenter and $C$-excenter of $ABC$, respectively. Let $Q$ be the reflection of $K$ with respect to $A$. Prove that the points $P,Q,I_B,I_C$ are concyclic.

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of integers numbers. Let $\Delta^1a_n=a_{n+1}-a_n$ for a non-negative integer $n$. Define $\Delta^Ma_n= \Delta^{M-1}a_{n+1}- \Delta^{M-1}a_n$. A sequence is patriota if there are positive integers $k,l$ such that $a_{n+k}=\Delta^Ma_{n+l}$ for all non-negative integers $n$. Determine, with proof, whether exists a sequence that the last value of $M$ for which the sequence is patriota is $2022$.

Initially, a natural number $n$ is written on the blackboard. Then, at each minute, Neymar chooses a divisor $d>1$ of $n$, erases $n$, and writes $n+d$. If the initial number on the board is $2022$, what is the largest composite number that Neymar will never be able to write on the blackboard?

Let $n$ be a positive integer number. Define $S(n)$ to be the least positive integer such that $S(n) \equiv n \pmod{2}$, $S(n) \geq n$, and such that there are not positive integers numbers $k,x_1,x_2,...,x_k$ such that $n=x_1+x_2+...+x_k$ and $S(n)=x_1^2+x_2^2+...+x_k^2$. Prove that there exists a real constant $c>0$ and a positive integer $n_0$ such that, for all $n \geq n_0$, $S(n) \geq cn^{\frac{3}{2}}$.

Some cells of a $10 \times 10$ are colored blue. A set of six cells is called gremista when the cells are the intersection of three rows and two columns, or two rows and three columns, and are painted blue. Determine the greatest value of $n$ for which it is possible to color $n$ chessboard cells blue such that there is not a gremista set.

A single player game has the following rules: initially, there are $10$ piles of stones with $1,2,...,10$ stones, respectively. A movement consists on making one of the following operations: i) to choose $2$ piles, both of them with at least $2$ stones, combine them and then add $2$ stones to the new pile; ii) to choose a pile with at least $4$ stones, remove $2$ stones from it, and then split it into two piles with amount of piles to be chosen by the player. The game continues until is not possible to make an operation. a) Give an example of a sequence of moves leading to the end of the game. b) Make a table with the total number of stones and the number of piles before and after the first 5 operations in your example above. c) Show that the number of piles with one stone in the end of the game is always the same, no matter how the movements are made.

The nonzero real numbers $a, b, c$ satisfy the following system: $$\begin{cases} a+ab=c\\ b+bc=a\\ c+ca=b \end{cases}$$Find all possible values of the $abc$.

Let $ABC$ be a triangle with incenter $I$ and let $\Gamma$ be its circumcircle. Let $M$ be the midpoint of $BC$, $K$ the midpoint of the arc $BC$ which does not contain $A$, $L$ the midpoint of the arc $BC$ which contains $A$ and $J$ the reflection of $I$ by the line $KL$. The line $LJ$ intersects $\Gamma$ again at the point $T\neq L$. The line $TM$ intersects $\Gamma$ again at the point $S\neq T$. Prove that $S, I, M, K$ lie on the same circle.

Let $ABC$ a triangle with $AB=BC$ and incircle $\omega$. Let $M$ the mindpoint of $BC$; $P, Q$ points in the sides $AB, AC$ such that $PQ\parallel BC$, $PQ$ is tangent to $\omega$ and $\angle CQM=\angle PQM$. Find the perimeter of triangle $ABC$ knowing that $AQ=1$.

Initially, a natural number $n$ is written on the blackboard. Then, at each minute, Esmeralda chooses a divisor $d>1$ of $n$, erases $n$, and writes $n+d$. If the initial number on the board is $2022$, what is the largest composite number that Esmeralda will never be able to write on the blackboard?

Determine the largest positive integer $k$ for which the following statement is true: given $k$ distinct subsets of the set $\{1, 2, 3, \dots , 2023\}$, each with $1011$ elements, it is possible partition the subsets into two collections so that any two subsets in one same collection have some element in common.