2

Part of 1994 Irish Math Olympiad

Problems(2)

equilateral triangle

Source: Ireland 1994

A,B,C

be collinear points on the plane with

B

A

C

. Equilateral triangles

ABD,BCE,CAF

are constructed with

D,E

on one side of the line

AC

F

on the other side. Prove that the centroids of the triangles are the vertices of an equilateral triangle, and that the centroid of this triangle lies on the line

AC

geometrycircumcircletrapezoidgeometric transformationrotationperpendicular bisectorgeometry proposed

find all possible values

Source: Ireland 1994

p,q,r

be distinct real numbers that satisfy: q\equal{}p(4\minus{}p), \, r\equal{}q(4\minus{}q), \, p\equal{}r(4\minus{}r). Find all possible values of p\plus{}q\plus{}r.

algebrapolynomial