2009 Flanders Math Olympiad

Part of Flanders Math Olympiad

Subcontests

(4)

1

2009 Belgians handshaking

In an attempt to beat the Belgian handshake record come on

20/09/2009

2009

Belgians together in a large sports hall. Among them are Nathalie and thomas. During this event, everyone shakes hands with everyone exactly once other attendees. Afterwards, Nathalie says: “I have exactly

5

times as many Flemish people shaken hands as people from Brussels.” Thomas replies with “I have exactly

3

times as much Walloons and Brussels people shook hands”. From which region does Nathalie come and from which region comes Thomas?

1

max no of solid regular tetrahedrons

The maximum number of solid regular tetrahedrons can be placed against each other so that one of their edges coincides with a given line segment in space?
[hide=original wording]Hoeveel massieve regelmatige viervlakken kan men maximaal tegen mekaar plaatsen zodat ´e´en van hun ribben samenvalt met een gegeven lijnstuk in de ruimte?

1

natural with 4 divisors wanted

A natural number has four natural divisors:

1

, the number itself, and two real divisors. That number plus

9

is equal to seven times the sum of the true divisors. Determine that number and prove that it is unique.

1

perpencicular bisector, angle bisector, moving points (2009 VWO Flanders MO p3)

Consider a line segment

[AB]

M

and perpendicular bisector

m

. For each point

X \ne M

on m consider we are the intersection point

Y

BX

with the bisector from the angle

\angle BAX

X

M

, then approaches

Y

[AB]

. Which? https://cdn.artofproblemsolving.com/attachments/a/3/17d72a23011a9ec22deb20184717cc9c020a2b.png [hide=original wording]Beschouw een lijnstuk [AB] met midden M en middelloodlijn m. Voor elk punt X 6= M op m beschouwenwe het snijpunt Y van de rechte BX met de bissectrice van de hoek < BAX . Als X tot M nadert, dan nadert Y tot een punt van [AB]. Welk?