Let $ABC$ a triangle and $C$ it´s circuncircle. Let $D$ a point in arc $AB$ that not contain $A$, diferent of $B$ and $C$ such that $CD$ and $AB$ are not parallel. Let $E$ the intersection of $CD$ and $AB$ and $O$ the circumcircle of triangle $DBE$. Prove that the measure of $\angle OBE$ does not depend of the choice of $D$.

In the Cancun´s league participate $30$ teams. For this tournament we want to divide the $30$ teams in $2$ groups such that: $\textbf{1.}$ Every team plays exactly $82$ games $\textbf{2.}$ The number of gamen between teams of differents groups is equal to the half of games played. Can we do this?

Let $p$ of prime of the form $3k+2$ such that $a^2+ab+b^2$ is divisible by $p$ for some integers $a$ and $b$. Prove that both of $a$ and $b$ are divisible by $p$.

Find all couples of positive integers $m$ and $n$ such that $$n!+5=m^3$$

Consider an acutangle triangle $ABC$ with circumcenter $O$. A circumference that passes through $B$ and $O$ intersect sides $BC$ and $AB$ in points $P$ and $Q$. Prove that the orthocenter of triangle $OPQ$ is on $AC$.

Consider $f_1=1, f_2=1$ and $f_{n+1}=f_n+f_{n-1}$ for $n\geq 2$. Determine if exists $n\leq 1000001$ such that the last three digits of $f_n$ are zero.