OMM 2003

Find all positive integers with two or more digits such that if we insert a $0$ between the units and tens digits we get a multiple of the original number.

$A, B, C$ are collinear with $B$ betweeen $A$ and $C$ . $K_{1}$ is the circle with diameter $AB$ , and $K_{2}$ is the circle with diameter $BC$ . Another circle touches $AC$ at $B$ and meets $K_{1}$ again at $P$ and $K_{2}$ again at $Q$ . The line $PQ$ meets $K_{1}$ again at $R$ and $K_{2}$ again at $S$ . Show that the lines $AR$ and $CS$ meet on the perpendicular to $AC$ at $B$ .

At a party there are $n$ women and $n$ men. Each woman likes $r$ of the men, and each man likes $s$ of then women. For which $r$ and $s$ must there be a man and a woman who like each other?

The quadrilateral $ABCD$ has $AB$ parallel to $CD$ . $P$ is on the side $AB$ and $Q$ on the side $CD$ such that $\frac{AP}{PB}= \frac{DQ}{CQ}$ . M is the intersection of $AQ$ and $DP$ , and $N$ is the intersection of $PC$ and $QB$ . Find $MN$ in terms of $AB$ and $CD$ .

Some cards each have a pair of numbers written on them. There is just one card for each pair $(a,b)$ with $1 \leq a < b \leq 2003$ . Two players play the following game. Each removes a card in turn and writes the product $ab$ of its numbers on the blackboard. The first player who causes the greatest common divisor of the numbers on the blackboard to fall to $1$ loses. Which player has a winning strategy?

Given a positive integer $n$ , an allowed move is to form $2n+1$ or $3n+2$ . The set $S_{n}$ is the set of all numbers that can be obtained by a sequence of allowed moves starting with $n$ . For example, we can form $5 \rightarrow 11 \rightarrow 35$ so $5, 11$ and $35$ belong to $S_{5}$ . We call $m$ and $n$ compatible if $S_{m}$ and $S_{n}$ has a common element. Which members of $\{1, 2, 3, ... , 2002\}$ are compatible with $2003$ ?