2

Part of 2008 Germany Team Selection Test

Problems(4)

An isosceles trapezium with AB || CD

Source: VAIMO 2008, Problem 2

ABCD

be an isosceles trapezium with

AB \parallel{} CD

and \bar{BC} \equal{} \bar{AD}. The parallel to

AD

B

meets the perpendicular to

AD

D

X.

The line through

A

drawn which is parallel to

BD

meets the perpendicular to

BD

D

Y.

Prove that points

C,X,D

Y

lie on a common circle.

geometrytrapezoidcircumcirclegeometry unsolved

Adding an arbitrary new edge gives rise to a 3-clique

Source: AIMO 2008, TST 1, P2

(i) Determine the smallest number of edges which a graph of

n

nodes may have given that adding an arbitrary new edge would give rise to a 3-clique (3 nodes joined pairwise by edges). (ii) Determine the smallest number of edges which a graph of

n

nodes may have given that adding an arbitrary new edge would give rise to a 4-clique (4 nodes joined pairwise by edges).

graph theorycombinatorics unsolvedcombinatorics

Tracey baked a square cake whose surface is dissected

Source: AIMO 2008, TST 4, P2, Suggested by Christian Reiher

Tracey baked a square cake whose surface is dissected in a

10 \times 10

grid. In some of the fields she wants to put a strawberry such that for each four fields that compose a rectangle whose edges run in parallel to the edges of the cake boundary there is at least one strawberry. What is the minimum number of required strawberries?

geometryrectanglecombinatorics unsolvedcombinatorics

Inequality for circumcircle radius of the triangle XYZ

Source: AIMO 2008, TST 5, P2, Suggested by Gunther Vogel

For three points

X,Y,Z

R_{XYZ}

be the circumcircle radius of the triangle

XYZ.

ABC

is a triangle with incircle centre

I

then we have: \frac{1}{R_{ABI}} \plus{} \frac{1}{R_{BCI}} \plus{} \frac{1}{R_{CAI}} \leq \frac{1}{\bar{AI}} \plus{} \frac{1}{\bar{BI}} \plus{} \frac{1}{\bar{CI}}.

inequalitiesgeometrycircumcircletrigonometrygeometry unsolved