DE-BUNDESWETTBEWERB 2002

Planet Ypsilon has a calendar similar to ours: A year consists of $365$ days, and every month has $28$ , $30$ or $31$ days. Prove that on Planet Ypsilon, a year must have $12$ months.

Consider a $12$ -gon with sidelengths $1$ , $2$ , $3$ , $4$ , ..., $12$ . Prove that there are three consecutive sides in this $12$ -gon, whose lengths have a sum $> 20$ .

A pile of cards, numbered with $1$ , $2$ , ..., $n$ , is being shuffled. Afterwards, the following operation is repeatedly performed: If the uppermost card of the pile has the number $k$ , then we reverse the order of the $k$ uppermost cards.

Prove that, after finitely many executions of this operation, the card with the number $1$ will become the uppermost card of the pile.

We consider the sequences strictely increasing $(a_0,a_1,...)$ of naturals which have the following property : For every natural $n$ , there is exactly one representation of $n$ as $a_i+2a_j+4a_k$ , where $i,j,k$ can be equal.

Prove that there is exactly a such sequence and find $a_{2002}$