OMM 1990

How many paths are there from $A$ to the line $BC$ if the path does not go through any vertex twice and always moves to the left?

$ABC$ is a triangle with $\angle B = 90^o$ and altitude $BH$ . The inradii of $ABC, ABH, CBH$ are $r, r_1, r_2$ . Find a relation between them.

Show that $n^{n-1}-1$ is divisible by $(n-1)^2$ for $n > 2$ .

Find $0/1 + 1/1 + 0/2 + 1/2 + 2/2 + 0/3 + 1/3 + 2/3 + 3/3 + 0/4 + 1/4 + 2/4 + 3/4 + 4/4 + 0/5 + 1/5 + 2/5 + 3/5 + 4/5 + 5/5 + 0/6 + 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6$

Given $19$ points in the plane with integer coordinates, no three collinear, show that we can always find three points whose centroid has integer coordinates.

$ABC$ is a triangle with $\angle C = 90^o$ . $E$ is a point on $AC$ , and $F$ is the midpoint of $EC$ . $CH$ is an altitude. $I$ is the circumcenter of $AHE$ , and $G$ is the midpoint of $BC$ . Show that $ABC$ and $IGF$ are similar.