3

Part of 1991 China Team Selection Test

Problems(2)

number of good circles is n

Source: China TST 1991, problem 3

5

points are given in the plane, any three non-collinear and any four non-concyclic. If three points determine a circle that has one of the remaining points inside it and the other one outside it, then the circle is said to be good. Let the number of good circles be

n

; find all possible values of

n

invariantinductioncombinatorial geometrycombinatorics unsolvedcombinatorics

edges of a polyhedron are painted with red or yellow

Source: China TST 1991, problem 6

All edges of a polyhedron are painted with red or yellow. For an angle of a facet, if the edges determining it are of different colors, then the angle is called excentric. The excentricity of a vertex

A

S_A

, is defined as the number of excentric angles it has. Prove that there exist two vertices

B

C

S_B + S_C \leq 4

inequalitiesgeometry3D geometrytetrahedroncombinatorics unsolvedcombinatoricsgraph theory