Lalo and Sergio play in a regular polygon of $n\geq 4$ sides. In his turn, Lalo paints a diagonal or side of pink, and in his turn Sergio paint a diagonal or side of orange. Wins the game who achieve paint the three sides of a triangle with his color, if none of the players can win, they game tie. Lalo starts playing. Determines all natural numbers $n$ such that one of the players have winning strategy.

Let $n=\frac{2^{2018}-1}{3}$. Prove that $n$ divides $2^n-2$.

Let $ABC$ a triangle with circumcircle $\Gamma$ and $R$ a point inside $ABC$ such that $\angle ABR=\angle RBC$. Let $\Gamma_1$ and $\Gamma_2$ the circumcircles of triangles $ARB$ and $CRB$ respectly. The parallel to $AC$ that pass through $R$, intersect $\Gamma$ in $D$ and $E$, with $D$ on the same side of $BR$ that $A$ and $E$ on the same side of $BR$ that $C$. $AD$ intersect $\Gamma_1$ in $P$ and $CE$ intersect $\Gamma_2$ in $Q$. Prove that $APQC$ is cyclic if and only if $AB=BC$

For every natural $n$ let $a_n=20\dots 018$ with $n$ ceros, for example, $a_1=2018, a_3=200018, a_7=2000000018$. Prove that there are infinity values of $n$ such that $2018$ divides $a_n$

Let $ABC$ an isosceles triangle with $CA=CB$ and $\Gamma$ it´s circumcircle. The perpendicular to $CB$ through $B$ intersect $\Gamma$ in points $B$ and $E$. The parallel to $BC$ through $A$ intersect $\Gamma$ in points $A$ and $D$. Let $F$ the intersection of $ED$ and $BC, I$ the intersection of $BD$ and $EC, \Omega$ the cricumcircle of the triangle $ADI$ and $\Phi$ the circumcircle of $BEF$.If $O$ and $P$ are the centers of $\Gamma$ and $\Phi$, respectively, prove that $OP$ is tangent to $\Omega$

Find all polynomials $p(x)$ such that for all reals $a, b$ and $c$, with $a+b+c=0$, satisfies $$p(a^3)+p(b^3)+p(c^3)=3p(abc)$$