4

Part of 1974 Bundeswettbewerb Mathematik

Problems(2)

Bundeswettbewerb Mathematik 1974 Problem 1.4

Source: Bundeswettbewerb Mathematik 1974 Round 1

All diagonals of a convex polygon are drawn. Prove that its sides and diagonals can be assigned arrows in such a way that no round trip along sides and diagonals is possible.

polygoncombinatoricscycles

Bundeswettbewerb Mathematik 1974 Problem 2.4

Source: Bundeswettbewerb Mathematik 1974 Round 2

Peter and Paul gamble as follows. For each natural number, successively, they determine its largest odd divisor and compute its remainder when divided by

4

. If this remainder is

1

, then Peter gives Paul a coin; otherwise, Paul gives Peter a coin. After some time they stop playing and balance the accounts. Prove that Paul wins.

number theorydivisoroddremainder