1

Part of 1986 Bundeswettbewerb Mathematik

Problems(2)

ways that polyline be segments of n points on circle doesn't cut itself

Source: 1986 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p1

n

points on a circle (

n > 1

). Denote them with

P_1,P_2, P_3, ..., P_n

such that the polyline

P_1P_2P_3... P_n

does not intersect itself. In how many ways is this possible?

combinatoricscombinatorial geometry

numbering edges of cube from 1 to 12

Source: 1986 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p1

The edges of a cube are numbered from

1

12

, then is calculated for each vertex the sum of the numbers of the edges going out from it. a) Prove that these sums cannot all be the same. b) Can eight equal sums result after one of the edge numbers is replaced by the number

13