In the country called Orientation a one-way traffic system was introduced on all the roads, and each city can be reached from any other one by driving on no more than two roads. One road was closed for repairs but from every city it remained possible to get to any other. Prove that for every two cities this can still be done whilst driving on no more than 3 roads.
In a circle, each member has one friend and one enemy. Prove that
a) the number of members is even.
b) the circle can be divided into two neutral circles.
Out of a whole 100-vertex graph, 98 edges were removed. Prove that the remaining ones were connected.
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.
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 \(A\).
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.
How many times greater is the sum of the numbers in the hundred and first line of the Pascal triangle than the sum of the numbers in the hundredth line?
Let’s put plus and minus signs in the 99th line of Pascal’s triangle. Between the first and second number there is a minus sign, between the second and the third there is a plus sign, between the third and the fourth there is a minus sign, then again a plus sign, and so on. Find the value of the resulting expression.