Problems

At the elections to the High Government every voter who comes, votes for himself (if he is a candidate) and for those candidates who are his friends. The forecast of the media service of the mayor’s office is considered good if it correctly predicts the number of votes at least for one of the candidates, and not considered good otherwise. Prove that for any forecast voters can show up at the elections in such a way that this forecast will not be considered good.

Prove Sperner’s lemma in dimension \(1\), namely on a line.
The simplex in this case is just a segment, the triangulation is subdivision of the segment into multiple small segments, and the conditions of a Sperner’s coloring are the following:

• There are only two colors;

• The opposite ends of the main segment are colored differently;

Then one needs to prove that there exists a small segment with two ends colored in different colors. In particular there is an odd number of such small segments.

Draw a Sperner’s coloring for the following \(3\)-dimensional simplex. The point \(F\) is on the face \(ABC\), point \(E\) is on the face \(BCD\), point \(G\) is on the face \(ACD\) and the point \(H\) is on the face \(ABD\).