Iranian Mathematical Olympiad

Question

A regular dodecahedron is a convex polyhedron such that its faces are regular pentagons. It has 20 vertices and 3 edges connected to each vertex. (As you see in the picture.)

Suppose that we have marked 10 vertices of a regular dodecahedron. a) Prove that we can rotate the dodecahedron in such a way that the dodecahedron is mapped to itself and at most four marked vertices are mapped to a marked vertex. b) Prove that number 4 cannot be replaced with number 3 in the previous part.

Accepted Answer

a) Now consider one of the marked vertices named

A

. We have 10 marked vertices and each of them can lie on its locations in 3 distinct ways, so in

3 \times 10 - 1 = 29

rotations (other than the original state) a special vertex lie on the location of

A

in original state, therefore totally

29 \times 10 = 290

times a marked vertex lie on a marked vertex of the original state. Now by the Pigeonhole principle less than

\frac{290}{59} < 5

of this coincidences is in one of the states and so we're done. □

b) It suffices to give an example that in each rotation at least 4 marked vertices lie on a marked vertex of original state. Mark the vertices of two opposite faces. Suppose that there exists a rotation with at most 3 coincidences. Consider this state is

ξ

. So one of the marked faces in

ξ

has at most one marked vertex of the original state, but each face has at least two marked vertices, contradiction. □