The Central American Olympiad is an annual competition. The ninth Olympiad is held in 2007. Find all the positive integers $n$ such that $n$ divides the number of the year in which the $n$-th Olympiad takes place.

In a triangle $ABC$, the angle bisector of $A$ and the cevians $BD$ and $CE$ concur at a point $P$ inside the triangle. Show that the quadrilateral $ADPE$ has an incircle if and only if $AB=AC$.

Let $S$ be a finite set of integers. Suppose that for every two different elements of $S$, $p$ and $q$, there exist not necessarily distinct integers $a \neq 0$, $b$, $c$ belonging to $S$, such that $p$ and $q$ are the roots of the polynomial $ax^{2}+bx+c$. Determine the maximum number of elements that $S$ can have.

In a remote island, a language in which every word can be written using only the letters $a$, $b$, $c$, $d$, $e$, $f$, $g$ is spoken. Let's say two words are synonymous if we can transform one into the other according to the following rules: i) Change a letter by another two in the following way: \[a \rightarrow bc,\ b \rightarrow cd,\ c \rightarrow de,\ d \rightarrow ef,\ e \rightarrow fg,\ f\rightarrow ga,\ g\rightarrow ab\] ii) If a letter is between other two equal letters, these can be removed. For example, $dfd \rightarrow f$. Show that all words in this language are synonymous.

Given two non-negative integers $m>n$, let's say that $m$ ends in $n$ if we can get $n$ by erasing some digits (from left to right) in the decimal representation of $m$. For example, 329 ends in 29, and also in 9. Determine how many three-digit numbers end in the product of their digits.

Consider a circle $S$, and a point $P$ outside it. The tangent lines from $P$ meet $S$ at $A$ and $B$, respectively. Let $M$ be the midpoint of $AB$. The perpendicular bisector of $AM$ meets $S$ in a point $C$ lying inside the triangle $ABP$. $AC$ intersects $PM$ at $G$, and $PM$ meets $S$ in a point $D$ lying outside the triangle $ABP$. If $BD$ is parallel to $AC$, show that $G$ is the centroid of the triangle $ABP$. Arnoldo Aguilar (El Salvador)