OBM 1999

Let $ABCDE$ be a regular pentagon. The star $ACEBD$ has area 1. $AC$ and $BE$ meet at $P$ , while $BD$ and $CE$ meet at $Q$ . Find the area of $APQD$ .

Show that, if $\sqrt{2}$ is written in decimal notation, there is at least one nonzero digit at the interval of 1,000,000-th and 3,000,000-th digits.

How many coins can be placed on a $10 \times 10$ board (each at the center of its square, at most one per square) so that no four coins form a rectangle with sides parallel to the sides of the board?

On planet Zork there are some cities. For every city there is a city at the diametrically opposite point. Certain roads join the cities on Zork. If there is a road between cities $P$ and $Q$ , then there is also a road between the cities $P'$ and $Q'$ diametrically opposite to $P$ and $Q$ . In plus, the roads do not cross each other and for any two cities $P$ and $Q$ it is possible to travel from $P$ to $Q$ .

The prices of Kriptonita in Urghs (the planetary currency) in two towns connected by a road differ by at most 100. Prove that there exist two diametrically opposite cities in which the prices of Kriptonita differ by at most 100 Urghs.

There are $n$ football teams in Tumbolia. A championship is to be organised in which each team plays against every other team exactly once. Ever match takes place on a sunday and each team plays at most one match each sunday. Find the least possible positive integer $m_n$ for which it is possible to set up a championship lasting $m_n$ sundays.

Given any triangle $ABC$ , show how to construct $A'$ on the side $AB$ , $B'$ on the side $BC$ and $C'$ on the side $CA$ , such that $ABC$ and $A'B'C'$ are similar (with $\angle A = \angle A', \angle B = \angle B', \angle C = \angle C'$ ) and $A'B'C'$ has the least possible area.