OBM 1980

Box $A$ contains black balls and box $B$ contains white balls. Take a certain number of balls from $A$ and place them in $B$ . Then take the same number of balls from $B$ and place them in $A$ . Is the number of white balls in $A$ then greater, equal to, or less than the number of black balls in $B$ ?

Show that for any positive integer $n > 2$ we can find $n$ distinct positive integers such that the sum of their reciprocals is $1$ .

Given a triangle $ABC$ and a point $P_0$ on the side $AB$ . Construct points $P_i, Q_i, R_i$ as follows. $Q_i$ is the foot of the perpendicular from $P_i$ to $BC, R_i$ is the foot of the perpendicular from $Q_i$ to $AC$ and $P_i$ is the foot of the perpendicular from $R_{i-1}$ to $AB$ . Show that the points $P_i$ converge to a point $P$ on $AB$ and show how to construct $P$ .

Given $5$ points of a sphere radius $r$ , show that two of the points are a distance $\le r \sqrt2$ apart.