6

Part of 2006 Iran Team Selection Test

Problems(2)

Source: Iran TST 2006

G

be a tournoment such that it's edges are colored either red or blue. Prove that there exists a vertex of

G

v

with the property that, for every other vertex

u

there is a mono-color directed path from

v

u

inductioncombinatorics proposedcombinatorics

Source: Iran TST 2006

Suppose we have a simple polygon (that is it does not intersect itself, but not necessarily convex). Show that this polygon has a diameter which is completely inside the polygon and the two arcs it creates on the polygon perimeter (the two arcs have 2 vertices in common) both have at least one third of the vertices of the polygon.

geometryperimeterinductionabsolute valuecombinatorics proposedcombinatorics