2

Part of 1984 Bundeswettbewerb Mathematik

Problems(2)

sum of squares of lengths of different segments of regular n-gon

Source: 1984 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p2

Given is a regular

n

-gon with circumradius

1

L

is the set of (different) lengths of all connecting segments of its endpoints. What is the sum of the squares of the elements of

L

geometrycombinatorial geometryregular polygon

subsets of plane wanted, at least 1 of 2 two semicircular arcs of AB

Source: 1984 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p2

Determine all bounded closed subsets

F

of the plane with the following property:

F

consists of at least two points and always contains two points

A

B

as well as at least one of the two semicircular arcs over the segment

AB

.
Definitions: A subset of the

F

of the plane is said to be closed if: For every point

P

of the plane that is not an element of

F

, there is a (non-degenerate) disc with center

P

that has no elements of

F

geometrycombinatorial geometry