2

Part of 1987 Bundeswettbewerb Mathematik

Problems(2)

Bundeswettbewerb Mathematik 1987 Problem 1.2

Source: Bundeswettbewerb Mathematik 1987 Round 1

n

be a positive integer and

M=\{1,2,\ldots, n\}.

T\subset M

is called heavy if each of its elements is greater or equal than

|T|.

f(n)

denote the number of heavy subsets of

M.

Describe a method for finding

f(n)

and use it to calculate

f(32).

combinatoricsSubsetcardinalitynumber

Bundeswettbewerb Mathematik 1987 Problem 2.2

Source: Bundeswettbewerb Mathematik 1987 Round 2

An arrow is assigned to each edge of a polyhedron such that for each vertex, there is an arrow pointing towards that vertex and an arrow pointing away from that vertex. Prove that there exist at least two faces such that the arrows on their boundaries form a cycle.

polyhedroncombinatoricsDirected graphscycles