3

Part of 1988 Vietnam National Olympiad

Problems(2)

Partitioned plane

Source: Vietnam NMO 1988 Problem 3

The plane is partitioned into congruent equilateral triangles such that any two of them which are not disjoint have either a common vertex or a common side. Is there a circle containing exactly 1988 points in its interior?

combinatorics unsolvedcombinatorics

Three pairwise skew lines in space

Source: Vietnam NMO 1988 Problem 6

a

b

c

be three pairwise skew lines in space. Prove that they have a common perpendicular if and only if

S_a \circ S_b \circ S_c

is a reflection in a line, where

S_x

denotes the reflection in line

x

geometrygeometric transformationreflectiongeometry unsolved