The country OMEC is divided in $5$ regions, each region is divided in $5$ districts, and, in each district, $1001$ people vote. Each person choose between $A$ or $B$. In a district, a candidate's letter wins if it's the letter with the most votes. In a region, a candidate's letter wins if it won in most districts. A candidate is the new president of OMEC if the candidate won in most regions. The candidate $A$ can rearrange the people of each district in each region (for example, A moves someone in District M to District N in region 1), but he can't change them to a different region. Find the minimum number of votes that the candidate $A$ needs to become the new president.

Find all pairs $(n, q)$ such that $n$ is a positive integer, $q$ is a not integer rational and $$n^q-q$$is an integer.

Let $ABC$ a triangle with circumcircle $\Gamma$ and circumcenter $O$. A point $X$, different from $A$, $B$, $C$, or their diametrically opposite points, on $\Gamma$, is chosen. Let $\omega$ the circumcircle of $COX$. Let $E$ the second intersection of $XA$ with $\omega$, $F$ the second intersection of $XB$ with $\omega$ and $D$ a point on line $AB$ such that $CD \perp EF$. Prove that $E$ is the circumcenter of $ADC$ and $F$ is the circumcenter of $BDC$.

Find all polynomials $P(x)$ such that, for all real numbers $x, y, z$ that satisfy $x+ y +z =0$, $$P(x) +P(y) +P(z)=0$$

In triangle $ABC$, $D$ is the middle point of side $BC$ and $M$ is a point on segment $AD$ such that $AM=3MD$. The barycenter of $ABC$ and $M$ are on the inscribed circumference of $ABC$. Prove that $AB+AC>3BC$.

A board $1$x$k$ is called guayaco if: -Each unit square is painted with exactly one of $k$ available colors. -If $gcd(i,k)>1$, the $i$th unit square is painted with the same color as $(i-1)$th unit square. -If $gcd(i, k)=1$, the $i$th unit square is painted with the same color as $(k-i)$th unit square. Sebastian chooses a positive integer $a$ and calculates the number of boards $1$x$a$ that are guayacos. After that, David chooses a positive integer $b$ and calculates the number of boards $1$x$b$ that are guayacos. David wins if the number of boards $1$x$a$ that are guayacos is the same as the number of boards $1$x$b$ that are guayacos, otherwise, Sebastian wins. Find all the pairs $(a,b) $ such that, with those numbers, David wins.