OIM 1990

Let $f$ be a function defined for the non-negative integers, such that:

a) $f(n)=0$ if $n=2^{j}-1$ for some $j \geq 0$ . b) $f(n+1)=f(n)-1$ otherwise.

i) Show that for every $n \geq 0$ there exists $k \geq 0$ such that $f(n)+n=2^{k}-1$ . ii) Find $f(2^{1990})$ .

Let $ABC$ be a triangle. $I$ is the incenter, and the incircle is tangent to $BC$ , $CA$ , $AB$ at $D$ , $E$ , $F$ , respectively. $P$ is the second point of intersection of $AD$ and the incircle. If $M$ is the midpoint of $EF$ , show that $P$ , $I$ , $M$ , $D$ are concyclic.

Let $b$ , $c$ be integer numbers, and define $f(x)=(x+b)^2-c$ .

i) If $p$ is a prime number such that $c$ is divisible by $p$ but not by $p^{2}$ , show that for every integer $n$ , $f(n)$ is not divisible by $p^{2}$ .

ii) Let $q \neq 2$ be a prime divisor of $c$ . If $q$ divides $f(n)$ for some integer $n$ , show that for every integer $r$ there exists an integer $n'$ such that $f(n')$ is divisible by $qr$ .

Let $\Gamma_{1}$ be a circle. $AB$ is a diameter, $\ell$ is the tangent at $B$ , and $M$ is a point on $\Gamma_{1}$ other than $A$ . $\Gamma_{2}$ is a circle tangent to $\ell$ , and also to $\Gamma_{1}$ at $M$ .

a) Determine the point of tangency $P$ of $\ell$ and $\Gamma_{2}$ and find the locus of the center of $\Gamma_{2}$ as $M$ varies.

b) Show that there exists a circle that is always orthogonal to $\Gamma_{2}$ , regardless of the position of $M$ .

$A$ and $B$ are two opposite vertices of an $n \times n$ board. Within each small square of the board, the diagonal parallel to $AB$ is drawn, so that the board is divided in $2n^{2}$ equal triangles. A coin moves from $A$ to $B$ along the grid, and for every segment of the grid that it visits, a seed is put in each triangle that contains the segment as a side. The path followed by the coin is such that no segment is visited more than once, and after the coins arrives at $B$ , there are exactly two seeds in each of the $2n^{2}$ triangles of the board. Determine all the values of $n$ for which such scenario is possible.

Let $f(x)$ be a cubic polynomial with rational coefficients. If the graph of $f(x)$ is tangent to the $x$ axis, prove that the roots of $f(x)$ are all rational.