OBM 2008

A positive integer is dapper if at least one of its multiples begins with $2008$ . For example, $7$ is dapper because $200858$ is a multiple of $7$ and begins with $2008$ . Observe that $200858 = 28694\times 7$ .

Prove that every positive integer is dapper.

Let $S$ be a set of $6n$ points in a line. Choose randomly $4n$ of these points and paint them blue; the other $2n$ points are painted green. Prove that there exists a line segment that contains exactly $3n$ points from $S$ , $2n$ of them blue and $n$ of them green.

Let $ABCD$ be a cyclic quadrilateral and $r$ and $s$ the lines obtained reflecting $AB$ with respect to the internal bisectors of $\angle CAD$ and $\angle CBD$ , respectively. If $P$ is the intersection of $r$ and $s$ and $O$ is the center of the circumscribed circle of $ABCD$ , prove that $OP$ is perpendicular to $CD$ .

Prove that for all integers $a > 1$ and $b > 1$ there exists a function $f$ from the positive integers to the positive integers such that $f(a\cdot f(n)) = b\cdot n$ for all $n$ positive integer.