In a summer camp that is going to last $n$ weeks, you want to divide the time into $3$ periods so that each period starts on a Monday and ends on a Sunday. The first period will be dedicated to artistic work, the second will be for sports and in the third there will be a technological workshop. During each term, a Monday will be chosen for an expert on the topic of the term to give a talk. Let $C(n)$ be the number of ways in which the activity calendar can be made. (For example, if $n=10$ one way the calendar could be done is by putting the first four weeks for art and the artist talk on the first Monday; the next $5$ weeks could be for sports, with the athlete visit on the fourth Monday of that period; the remaining week would be for the technology workshop and the talk would be on Monday of that week.) Calculate $C(8)$.

In the isosceles triangle $ABC$, with $AB=AC$, $D$ is a point on the extension of $CA$ such that $DB$ is perpendicular to $BC$, $E$ is a point on the extension of $BC$ such that $CE=2BC$, and $F$ is a point on $ED$ such that $FC$ is parallel to $AB$. Prove that $FA$ is parallel to $BC$.

On a circular board there are $19$ squares numbered in order from $1$ to $19$ (to the right of $1$ is $2$, to the right of it is $3$, and so on, until $1$ is to the right of $19$). In each box there is a token. Every minute each checker moves to its right the number of the box it is in at that moment plus one; for example, the piece that is in the $7$th place leaves the first minute $7 + 1$ places to its right until the $15$th square; the second minute that same checker moves to your right $15 + 1$ places, to square $12$, etc. Determine if at some point all the tokens reach the place where they started and, if so, say how many minutes must elapse. original wordingEn un tablero circular hay 19 casillas numeradas en orden del 1 al 19 (a la derecha del 1 está el 2, a la derecha de éste está el 3 y así sucesivamente, hasta el 1 que está a la derecha del 19). En cada casilla hay una ficha. Cada minuto cada ficha se mueve a su derecha el número de la casilla en que se encuentra en ese momento más una; por ejemplo, la ficha que está en el lugar 7 se va el primer minuto 7 + 1 lugares a su derecha hasta la casilla 15; el segundo minuto esa misma ficha se mueve a su derecha 15 + 1 lugares, hasta la casilla 12, etc. Determinar si en algún momento todas las fichas llegan al lugar donde empezaron y, si es así, decir cuántos minutos deben transcurrir.