3

Part of 1976 Bundeswettbewerb Mathematik

Problems(2)

Bundeswettbewerb Mathematik 1976 Problem 1.3

Source: Bundeswettbewerb Mathematik 1976 First round

S

of rational numbers is ordered in a tree-diagram in such a way that each rational number

\frac{a}{b}

a

b

are coprime integers) has exactly two successors:

\frac{a}{a+b}

\frac{b}{a+b}

. How should the initial element be selected such that this tree contains the set of all rationals

r

0 < r < 1

? Give a procedure for determining the level of a rational number

\frac{p}{q}

treerationalalgebra

Bundeswettbewerb Mathematik 1976 Problem 2.3

Source: Bundeswettbewerb Mathematik 1976 Second round

A circle is divided by

2n

2n

equal arcs. Let

P_1, P_2, \ldots, P_{2n}

be an arbitrary permutation of the

2n

division points. Prove that the polygonal line

P_1 P_2 \cdots P_{2n} P_1

contains at least two parallel segments.

circlecombinatoricsparallel