OMCC 2007

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.

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.

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)