Determine all triples $(x,y,p)$ of positive integers such that $p$ is prime, $p=x^2+1$ and $2p^2=y^2+1$.

Let $ABCD$ be a convex quadrilateral with $AB>AD$ and $\angle B=\angle D=90^{\circ}$. Let $P$ be a point in the side $AB$ such that $AP=AD$. The lines $PD$ and $BC$ cut in the point $Q$. The perpendicular line to $AC$ passing by $Q$ cuts $AB$ in the point $R$. Let $S$ be the foot of perpendicular of $D$ to the line $AC$. Prove that $\angle PSQ=\angle RCP$.

The water city of Platense consists of many platforms and bridges between them. Each bridge connects two platforms and there is not two bridges connecting the same two platforms. The mayor wants to switch some bridges by a series of moves in the following way: if there are three platforms $A,B,C$ and bridges $AB$ and $AC$ (no bridge $BC$), he can switch bridge $AB$ to a bridge $BC$. A configuration of bridges is good if it is possible to go to any platfom from any platform using only bridges. Starting in a good configuration, prove that the mayor can reach any other good configuration, whose the quantity of bridges is the same, using the allowed moves.

Luffy is playing with some magic boxes and a machine. Each box has a value(number) inside. Opening a box, Luffy sees the value, adds the value to his score(if the box value is negative, Luffy loses points) and destroys the box. Putting a box of value $X$ in the machine, this box vanishes and it is replaced by two new boxes of values $X+1$ and $X-1$(it's not known which one has the respective value, but he can identify the new boxes). At the beginning, Luffy has $0$ points and has a box whose value is known(it is zero). a) Prove that Luffy can reach at least $1000$ points b) Is it possible that Luffy reaches at least $1000000$ points, without have less than $-42$ points in any moment?

Let $\mathbb{R}^{+}$ be the set of positive real numbers. Determine all non-negative real number $\alpha$ such that there exist a function $f:\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $$f(x^{\alpha}+y)=(f(x+y))^{\alpha}+f(y)$$for any $x,y$ positive real numbers.

Let $ABC$ be an acute-angled triangle such that $AB+BC=4AC$. Let $D$ in $AC$ such that $BD$ is angle bisector of $\angle ABC$. In the segment $BD$, points $P$ and $Q$ are marked such that $BP=2DQ$. The perpendicular line to $BD$, passing by $Q$, cuts the segments $AB$ and $BC$ in $X$ and $Y$, respectively. Let $L$ be the parallel line to $AC$ passing by $P$. The point $B$ is in a different half-plane(with respect to the line $L$) of the points $X$ and $Y$. An ant starts a run in the point $X$, goes to a point in the line $AC$, after that goes to a point in the line $L$, returns to a point in the line $AC$ and finishes in the point $Y$. Prove that the least length of the ant's run is equal to $4XY$.

An integer $n\geq 3$ is poli-pythagorean if there exist $n$ positive integers pairwise distinct such that we can order these numbers in the vertices of a regular $n$-gon such that the sum of the squares of consecutive vertices is also a perfect square. For instance, $3$ is poli-pythagorean, because if we write $44,117,240$ in the vertices of a triangle we notice: $$44^2+117^2=125^2, 117^2+240^2=267^2, 240^2+44^2=244^2$$Determine all poli-pythagorean integers.

Let $ABCD$ be a convex quadrilateral, such that $AB = CD$, $\angle BCD = 2 \angle BAD$, $\angle ABC = 2 \angle ADC$ and $\angle BAD \neq \angle ADC$. Determine the measure of the angle between the diagonals $AC$ and $BD$.

Let $n>d>0$ integers. Batman, Joker, Clark play the following game in an infinite checkered board. Initially, Batman and Joker are in cells with distance $n$ and a candy is in a cell with distance $d$ to Batman. Batman is blindfold, and can only see his cell. Clark and Joker can see the whole board. The following two moves go alternately. 1 - Batman goes to an adjacent cell. If he touches Joker, Batman loses. If he touches the candy, Batman wins. If the cell is empty, Clark chooses to say loudly one of the following two words hot or cold. 2 - Joker goes to an adjacent cell. If he touches Batman or candy, Joker wins. Otherwise, the game continues. Determine for each $d$, the least $n$, such that Batman, and Clark can plan an strategy to ensure the Batman's win, regardless of initial positions of the Joker and of the candy. Note: Two cells are adjacent if its have a common side. The distance between two cells $X$ and $Y$ is the least $p$ such that there exist cells $X=X_0,X_1,X_2,\dots, X_p=Y$ with $X_i$ adjacent to $X_{i-1}$ for all $i=1,2,\dots,p$.

A set of points on the plane is antiparallelogram if any four points of the set are not vertices of a parallelogram. Prove that for any set of $2023$ points on the plane, no three of them are collinears, there exists a subset of $17$ points, such that this subset is antiparallelogram.

A positive integer $N$ is rioplatense if it satifies the following conditions: 1 -There exist $34$ consecutive integers such that its product is divisible by $N$, but none of them is divisible by $N$. 2 - There not exist $30$ consecutive integers such that its product is divisible by $N$, but none of them is divisible by $N$. Determine all rioplatense numbers.

Find all functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ such that $$f(x+f(y+1))+f(xy)=f(x+1)(f(y)+1)$$for any $x,y$ integers.

A group of $4046$ friends will play a videogame tournament. For that, $2023$ of them will go to one room which the computers are labeled with $a_1,a_2,\dots,a_{2023}$ and the other $2023$ friends go to another room which the computers are labeled with $b_1,b_2,\dots,b_{2023}$. The player of computer $a_i$ always challenges the players of computer $b_i,b_{i+2},b_{i+3},b_{i+4}$(the player doesn't challenge $b_{i+1}$). After the first round, inside both rooms, the players may switch the computers. After the reordering, all the players realize that they are challenging the same players of the first round. Prove that if one player has not switched his computer, then all the players have not switched their computers.