Problems

In a country, each two cities are connected with a one-way road.

Prove that there is a city from which you can drive to any other whilst travelling along no more than two roads.

a) What is the minimum number of pieces of wire needed in order to weld a cube’s frame?

b) What is the maximum length of a piece of wire that can be cut from this frame? (The length of the edge of the cube is 1 cm).

Prove that in a bipartite planar graph \(E \geq 2F\), if \(E \geq 2\) (\(E\) is the number of edges, \(F\) is the number of regions).

12 teams played a volleyball tournament in one round. Two teams scored exactly 7 wins.

Prove that there are teams \(A\), \(B\), \(C\) where \(A\) won against \(B\), \(B\) won against \(C\), and \(C\) won against \(A\).

On a plane, there are 1983 points and a circle of unit radius. Prove that there is a point on the circle, from which the sum of the distances to these points is no less than 1983.

Is it possible to draw five lines from one point on a plane so that there are exactly four acute angles among the angles formed by them? Angles between not only neighboring rays, but between any two rays, can be considered.

Is it possible to draw this picture (see the figure), without taking your pencil off the paper and going along each line only once?

A scone contains raisins and sultanas. Prove that inside the scone there will always be two points 1cm apart such that either both lie inside raisins, both inside sultanas, or both lie outside of either raisins or sultanas.

Does the number of 1999 occur in the Pascal triangle?

Find a natural number greater than one that occurs in the Pascal triangle a) more than three times; b) more than four times.