3

Part of 2014 Bundeswettbewerb Mathematik

Problems(2)

BWM 2014 P3: Construct with a ruler, given a regular hexagon

Source:

A regular hexagon with side length

1

is given. Using a ruler construct points in such a way that among the given and constructed points there are two such points that the distance between them is

\sqrt7

.
Notes: ''Using a ruler construct points

\ldots

'' means: Newly constructed points arise only as the intersection of straight lines connecting two points that are given or already constructed. In particular, no length can be measured by the ruler.

geometryparallelogram

Find the max number of points of intersection of semicircles

Source: P7: BWM 2014

g

is given in a plane.

n

distinct points are chosen arbitrarily from

g

and are named as

A_1, A_2, \ldots, A_n

. For each pair of points

A_i,A_j

, a semicircle is drawn with

A_i

A_j

as its endpoints. All semicircles lie on the same side of

g

. Determine the maximum number of points (which are not lying in

g

) of intersection of semicircles as a function of

n

functioncombinatorics proposedcombinatorics