1

Part of 2002 Bundeswettbewerb Mathematik

Problems(2)

Planet ypsilon and its calendar

Source: German Math. Competition BWM 2002, 1st round, problem 1

Planet Ypsilon has a calendar similar to ours: A year consists of

365

days, and every month has

28

30

31

days. Prove that on Planet Ypsilon, a year must have

12

The 1 will come up

Source: German Math. Competition BWM 2002, 2nd round, problem 1

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.

inductioncombinatorics proposedcombinatorics