OPM 2000

Determine, for each positive integer $n$ , the largest positive integer $k$ such that $2^k$ is a divisor of $3^n+1$ .

Calculates the sum of all numbers that can be formed using each of the odd digits once, that is, the numbers $13579$ , $13597$ , ..., $97531$ .

In the figure, $[ABC]$ and $[DEC]$ are right triangles . Knowing that $EB = 1/2, EC = 1$ and $AD = 1$ , calculate $DC$ .

In a tournament, $n$ players participate. Each player plays each other exactly once, with no ties. A player $A$ is said to be champion if, for every other player $B$ , one of the following two situations occurs: (a) $A$ beat $B$ ; (b) $A$ beat a player $C$ who in turn beat $B$ . Prove that in such a tournament there cannot be exactly two champions.