OBM 2006

Let $ABC$ be a triangle. The internal bisector of $\angle B$ meets $AC$ in $P$ and $I$ is the incenter of $ABC$ . Prove that if $AP+AB = CB$ , then $API$ is an isosceles triangle.

Let $n$ be an integer, $n \geq 3$ . Let $f(n)$ be the largest number of isosceles triangles whose vertices belong to some set of $n$ points in the plane without three colinear points. Prove that there exists positive real constants $a$ and $b$ such that $an^{2}< f(n) < bn^{2}$ for every integer $n$ , $n \geq 3$ .

Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that $$f(xf(y)+f(x)) = 2f(x)+xy$$ for every reals $x,y$ .

A positive integer is bold iff it has $8$ positive divisors that sum up to $3240$ . For example, $2006$ is bold because its $8$ positive divisors, $1$ , $2$ , $17$ , $34$ , $59$ , $118$ , $1003$ and $2006$ , sum up to $3240$ . Find the smallest positive bold number.

Let $P$ be a convex $2006$ -gon. The $1003$ diagonals connecting opposite vertices and the $1003$ lines connecting the midpoints of opposite sides are concurrent, that is, all $2006$ lines have a common point. Prove that the opposite sides of $P$ are parallel and congruent.

Professor Piraldo takes part in soccer matches with a lot of goals and judges a match in his own peculiar way. A match with score of $m$ goals to $n$ goals, $m\geq n$ , is tough when $m\leq f(n)$ , where $f(n)$ is defined by $f(0) = 0$ and, for $n \geq 1$ , $f(n) = 2n-f(r)+r$ , where $r$ is the largest integer such that $r < n$ and $f(r) \leq n$ .

Let $\phi ={1+\sqrt 5\over 2}$ . Prove that a match with score of $m$ goals to $n$ , $m\geq n$ , is tough if $m\leq \phi n$ and is not tough if $m \geq \phi n+1$ .