1

Part of 2007 Italy TST

Problems(2)

Locus of centers of rectangles

Source: Italian TST day 1, nÂ°1

ABC

an acute triangle. (a) Find the locus of points that are centers of rectangles whose vertices lie on the sides of

ABC

; (b) Determine if exist some points that are centers of

3

distinct rectangles whose vertices lie on the sides of

ABC

geometryrectangleincentergeometric transformationhomothetyreflectiongeometry proposed

An easy one about coloring graphs

Source: Italian TST , day 2, nÂ°1

We have a complete graph with

n

vertices. We have to color the vertices and the edges in a way such that: no two edges pointing to the same vertice are of the same color; a vertice and an edge pointing him are coloured in a different way. What is the minimum number of colors we need?

inequalitiesgraph theorycombinatorics proposedcombinatorics