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.
Four football teams play in a tournament. There’s the Ulams (\(U\)), the Vandermondes (\(V\)), the Wittgensteins (\(W\)) and the Xenos (\(X\)). Each team plays every other team
exactly once, and matches can end in a draw.
If a game ends in a draw, then both teams get \(1\) point. Otherwise, the winning team gets
\(3\) points and the losing team gets
\(0\) points. At the end of the
tournament, the teams have the following points totals: \(U\) has \(7\), \(V\)
has \(4\), \(W\) has \(3\) and \(X\) has \(2\).
Work out the results of each match, including showing that there’s no other way the results could have played out.