Let $ABCD$ be a quadrilateral, $E$ the midpoint of side $BC$, and $F$ the midpoint of side $AD$. Segment $AC$ intersects segment $BF$ at $M$ and segment $DE$ at $N$. If quadrilateral $MENF$ is also known to be a parallelogram, prove that $ABCD$ is also a parallelogram.

In the training of a state, the coach proposes a game. The coach writes four real numbers on the board in order from least to greatest: $a < b < c < d$. Each Olympian draws the figure on the right in her notebook and arranges the numbers inside the corner shapes, however she wants, putting a number on each one. Once arranged, on each segment write the square of the difference of the numbers at its ends. Then, add the $4$ numbers obtained. For example, if Vania arranges them as in the figure on the right, then the result would be $$ (c - b)^2 + (b- a)^2 + (a - d)^2 + (d - c)^2.$$ The Olympians with the lowest result win. In what ways can you arrange the numbers to win? Give all the possible solutions.

All the squares of a $2022 \times 2022$ board will be colored white or black. Chips will be placed in several of these boxes, at most one per box. We say that two tokens attack each other, when the following two conditions are met: a) There is a path of squares that joins the squares where the pieces were placed. This path can have a horizontal, vertical, or diagonal direction. b) All the squares in this path, including the squares where the pieces are, are of the same color. For example, the following figure shows a small example of a possible coloring of a $6 \times 6$ board with $A, B, C, D$, and $E$ tiles placed. The pairs of checkers that attack each other are $(D, E)$, $(C, D)$, and $(B, E)$. What is the maximum value of $k$ such that it is possible to color the board and place $k$ tiles without any two of them attacking each other?

Let $k$ be a positive integer and $m$ be an odd integer. Prove that there exists a positive integer $n$ such that $n^n-m$ is divisible by $2^k$.

A biologist found a pond with frogs. When classifying them by their mass, he noticed the following: The $50$ lightest frogs represented $30\%$ of the total mass of all the frogs in the pond, while the $44$ heaviest frogs represented $27\%$ of the total mass. As fate would have it, the frogs escaped and the biologist only has the above information. How many frogs were in the pond?

Let $a$ and $b$ be positive integers such that $$\frac{5a^4+a^2}{b^4+3b^2+4}$$is an integer. Prove that $a$ is not a prime number.

Let $ABCD$ be a parallelogram (non-rectangle) and $\Gamma$ is the circumcircle of $\triangle ABD$. The points $E$ and $F$ are the intersections of the lines $BC$ and $DC$ with $\Gamma$ respectively. Define $P=ED\cap BA$, $Q=FB\cap DA$ and $R=PQ\cap CA$. Prove that $$\frac{PR}{RQ}=(\frac{BC}{CD})^2$$

Let $n$ be a positive integer. Consider a figure of a equilateral triangle of side $n$ and splitted in $n^2$ small equilateral triangles of side $1$. One will mark some of the $1+2+\dots+(n+1)$ vertices of the small triangles, such that for every integer $k\geq 1$, there is not any trapezoid(trapezium), whose the sides are $(1,k,1,k+1)$, with all the vertices marked. Furthermore, there are no small triangle(side $1$) have your three vertices marked. Determine the greatest quantity of marked vertices. Note: The figure shows an example of a board when $n = 4$. and an example of one of the mentioned trapezoids, with three of its vertices marked and with $k = 2$.

Determine all finite nonempty sets $S$ of positive integers satisfying \[ {i+j\over (i,j)}\qquad\mbox{is an element of S for all i,j in S}, \] where $(i,j)$ is the greatest common divisor of $i$ and $j$.

Consider $\triangle ABC$ an isosceles triangle such that $AB = BC$. Let $P$ be a point satisfying $$\angle ABP = 80^\circ, \angle CBP = 20^\circ, \textrm{and} \hspace{0.17cm} AC = BP$$ Find all possible values of $\angle BCP$.

Consider a set $S$ of $16$ lattice points. The $16$ points of $S$ are divided into $8$ pairs in such a way that for every point $A$ and any of the $7$ pairs of points $(B,C)$ where $A$ is not included, $A$ is at a distance of at most $\sqrt{5}$ from either $B$ or $C$ Prove that any two points in the set $S$ are at a distance of at most $3\sqrt5$.