Let $C_1$ and $C_2$ be two circles with centers $O_1$ and $O_2$, respectively, intersecting at $A$ and $B$. Let $l_1$ be the line tangent to $C_1$ passing trough $A$, and $l_2$ the line tangent to $C_2$ passing through $B$. Suppose that $l_1$ and $l_2$ intersect at $P$ and $l_1$ intersects $C_2$ again at $Q$. Show that $PO_1B$ and $PO_2Q$ are similar triangles. Proposed by Pablo Valeriano

Find all pairs of positive integers $(m, n)$ such that $m^2-mn+n^2+1$ divides both numbers $3^{m+n}+(m+n)!$ and $3^{m^3+n^3}+m+n$. Proposed by Dorlir Ahmeti

Let $\mathbb{Z}$ be the set of integers. Find all functions $f: \mathbb{Z}\rightarrow \mathbb{Z}$ such that, for any two integers $m, n$, $$f(m^2)+f(mf(n))=f(m+n)f(m).$$ Proposed by Victor Domínguez and Pablo Valeriano

Find all positive integers $n$ such that it is possible to split the numbers from $1$ to $2n$ in two groups $(a_1,a_2,..,a_n)$, $(b_1,b_2,...,b_n)$ in such a way that $2n\mid a_1a_2\cdots a_n+b_1b_2\cdots b_n-1$. Proposed by Alef Pineda

There are $n\geq 2$ people at a party. Each person has at least one friend inside the party. Show that it is possible to choose a group of no more than $\frac{n}{2}$ people at the party, such that any other person outside the group has a friend inside it.

Let $ABC$ be a scalene triangle with circumcenter $O$, and let $D$ and $E$ be points inside angle $\measuredangle BAC$ such that $A$ lies on line $DE$, and $\angle ADB=\angle CBA$ and $\angle AEC=\angle BCA$. Let $M$ be the midpoint of $BC$ and $K$ be a point such that $OK$ is perpendicular to $AO$ and $\angle BAK=\angle MAC$. Finally, let $P$ be the intersection of the perpendicular bisectors of $BD$ and $CE$. Show that $KO=KP$. Proposed by Victor Domínguez