In a group of friends, each two people have exactly five common acquaintances. Prove that the number of pairs of friends is divisible by 3.
Find the largest value of the expression \(a + b + c + d - ab - bc - cd - da\), if each of the numbers \(a\), \(b\), \(c\) and \(d\) belongs to the interval \([0, 1]\).
At a contest named “Ah well, monsters!”, 15 dragons stand in a row. Between neighbouring dragons the number of heads differs by 1. If the dragon has more heads than both of his two neighbors, he is considered cunning, if he has less than both of his neighbors – strong, the rest (including those standing at the edges) are considered ordinary. In the row there are exactly four cunning dragons – with 4, 6, 7 and 7 heads and exactly three strong ones – with 3, 3 and 6 heads. The first and last dragons have the same number of heads.
a) Give an example of how this could occur.
b) Prove that the number of heads of the first dragon in all potential examples is the same.
Airlines connect pairs of cities. How can you connect 50 cities with the fewest number of airlines so that from every city you can get to any other city by taking at most two flights?