Let $p$ be an odd prime number. Let $S=a_1,a_2,\dots$ be the sequence defined as follows: $a_1=1,a_2=2,\dots,a_{p-1}=p-1$, and for $n\ge p$, $a_n$ is the smallest integer greater than $a_{n-1}$ such that in $a_1,a_2,\dots,a_n$ there are no arithmetic progressions of length $p$. We say that a positive integer is a ghost if it doesn’t appear in $S$. What is the smallest ghost that is not a multiple of $p$? Proposed by Guerrero

The Mictlán is an $n\times n$ board and each border of each $1\times 1$ cell is painted either purple or orange. Initially, a catrina lies inside a $1\times 1$ cell and may move in four directions (up, down, left, right) into another cell with the condition that she may move from one cell to another only if the border that joins them is painted orange. We know that no matter which cell the catrina chooses as a starting point, she may reach all other cells through a sequence of valid movements and she may not abandon the Mictlán (she may not leave the board). What is the maximum number of borders that could have been colored purple? Proposed by CDMX

Let $W,X,Y$ and $Z$ be points on a circumference $\omega$ with center $O$, in that order, such that $WY$ is perpendicular to $XZ$; $T$ is their intersection. $ABCD$ is the convex quadrilateral such that $W,X,Y$ and $Z$ are the tangency points of $\omega$ with segments $AB,BC,CD$ and $DA$ respectively. The perpendicular lines to $OA$ and $OB$ through $A$ and $B$, respectively, intersect at $P$; the perpendicular lines to $OB$ and $OC$ through $B$ and $C$, respectively, intersect at $Q$, and the perpendicular lines to $OC$ and $OD$ through $C$ and $D$, respectively, intersect at $R$. $O_1$ is the circumcenter of triangle $PQR$. Prove that $T,O$ and $O_1$ are collinear. Proposed by CDMX

Two types of pieces, bishops and rooks, are to be placed on a $10\times 10$ chessboard (without necessarily filling it) such that each piece occupies exactly one square of the board. A bishop $B$ is said to attack a piece $P$ if $B$ and $P$ are on the same diagonal and there are no pieces between $B$ and $P$ on that diagonal; a rook $R$ is said to attack a piece $P$ if $R$ and $P$ are on the same row or column and there are no pieces between $R$ and $P$ on that row or column. A piece $P$ is chocolate if no other piece $Q$ attacks $P$. What is the maximum number of chocolate pieces there may be, after placing some pieces on the chessboard? Proposed by José Alejandro Reyes González

Let $ABCD$ be a parallelogram. Half-circles $\omega_1,\omega_2,\omega_3$ and $\omega_4$ with diameters $AB,BC,CD$ and $DA$, respectively, are erected on the exterior of $ABCD$. Line $l_1$ is parallel to $BC$ and cuts $\omega_1$ at $X$, segment $AB$ at $P$, segment $CD$ at $R$ and $\omega_3$ at $Z$. Line $l_2$ is parallel to $AB$ and cuts $\omega_2$ at $Y$, segment $BC$ at $Q$, segment $DA$ at $S$ and $\omega_4$ at $W$. If $XP\cdot RZ=YQ\cdot SW$, prove that $PQRS$ is cyclic. Proposed by José Alejandro Reyes González

The sequence $a_1,a_2,\dots$ of positive integers obeys the following two conditions: For all positive integers $m,n$, it happens that $a_m\cdot a_n=a_{mn}$ There exist infinite positive integers $n$ such that $(a_1,a_2,\dots,a_n)$ is a permutation of $(1,2,\dots,n)$ Prove that $a_n=n$ for all positive integers $n$. Proposed by José Alejandro Reyes González