3

Part of 2005 Argentina Team Selection Test

Problems(2)

Equilaterals outside ABC

Source: Argentina IMO 2005 TST, problem 3

Given the triangle

ABC

we consider the points

X,Y,Z

such that the triangles

ABZ,BCX,CAZ

are equilateral, and they don't have intersection with

ABC

B'

be the midpoint of

BC

N'

the midpoint of

CY

M,N

the midpoints of

AZ,CX

, respectively. Prove that

B'N' \bot MN

ratiogeometrygeometric transformationrotationpower of a pointradical axisgeometry proposed

Graph problem, not so easy

Source: Argentina IMO 2005 TST, problem 6

We say that a group of

k

n-acceptable

if removing any boy from the group one can always find, in the other

k-1

group, a group of

n

boys such that everyone knows each other. For each

n

, find the biggest

k

such that in any group of

k

n-acceptable

we must always have a group of

n+1

boys such that everyone knows each other.

combinatorics proposedcombinatorics