Maria have $14$ days to train for an olympiad. The only conditions are that she cannot train by $3$ consecutive days and she cannot rest by $3$ consecutive days. Determine how many configurations of days(in training) she can reach her goal.

Let $ABC$ be a triangle, the point $E$ is in the segment $AC$, the point $F$ is in the segment $AB$ and $P=BE\cap CF$. Let $D$ be a point such that $AEDF$ is a parallelogram, Prove that $D$ is in the side $BC$, if and only if, the triangle $BPC$ and the quadrilateral $AEPF$ have the same area.

Let $a_0,a_1,a_2,\dots$ be a periodic sequence of real numbers(that is, there is a fixed positive integer $k$ such that $a_n=a_{n+k}$ for every integer $n\geq 0$). The following equality is true, for all $n\geq 0$: $a_{n+2}=\frac{1}{n+2} (a_n - \frac{n+1}{a_{n+1}})$ if $a_0=2020$, determine the value of $a_1$.

Determine all positive integers $n$ such that $\frac{n(n-1)}{2}-1$ divides $1^7+2^7+\dots +n^7$.

Determine the quantity of positive integers $N$ of $10$ digits with the following properties: I- All the digits of $N$ are non-zero. II- $11|N$. III- $N$ and all the permutation(s) of the digits of $N$ are divisible by $12$.

Prove that for each positive integer $n$, there exists a number $M$, such that $M$ can be written as sum of $1,2,3,\dots, n$ distinct perfect squares.

Between the states of Alinaesquina and Berlinda, each road connects one city of Alinaesquina to one city of Berlinda. All the roads are in two-ways, and between any two cities, it is possible to travel from one to the other, using only these (possibly more than one) roads. Furthermore, it is known that, from any city of anyone of the two states, the same number of $k$ roads are going out. We know that $k\geq 2$. Prove that governments of the states can close anyone of the roads, and there will still be a route (possibly through several roads) between any two cities.

Let $ABC$ be a triangle and $D$ is a point inside of $\triangle ABC$. The point $A'$ is the midpoint of the arc $BDC$, in the circle which passes by $B,C,D$. Analogously define $B'$ and $C'$, being the midpoints of the arc $ADC$ and $ADB$ respectively. Prove that the four points $D,A',B',C'$ are concyclic.

Let $D$ and $E$ be points on sides $AB$ and $AC$ of a triangle $ABC$ such that $DB = BC = CE$. The segments $BE$ and $CD$ intersect at point $P$. Prove that the incenter of triangle $ABC$ lies on the circles circumscribed around the triangles $BDP$ and $CEP$.

A number n is called charming when $ 4k^2 + n $ is a prime number for every $ 0 \leq k <n $ integer, find all charming numbers.

Let $a_1,a_2, \cdots$ be a sequence of integers that satisfies: $a_1=1$ and $a_{n+1}=a_n+a_{\lfloor \sqrt{n} \rfloor} , \forall n\geq 1 $. Prove that for all positive $k$, there is $m \geq 1$ such that $k \mid a_m$.

A flea is, initially, in the point, which the coordinate is $1$, in the real line. At each second, from the coordinate $a$, the flea can jump to the coordinate point $a+2$ or to the coordinate point $\frac{a}{2}$. Determine the quantity of distinct positions(including the initial position) which the flea can be in until $n$ seconds. For instance, if $n=1$, the flea can be in the coordinate points $1,3$ or $\frac{1}{2}$.