Sixty children came to a maths circle at UCL. Among any ten children who came to the circle there are three from the same school. Show that there are 15 children from the same school among all the children who came to the maths circle.
The people in Wonderland are having an election. Every voter writes 10 candidate names on a bulletin and puts it in a ballot box.
There are 11 ballot boxes all together. The March Hare, who is counting the votes, is very surprised to discover that there is at least one bulletin in each ballot box. Moreover, he learned that if he takes one bulletin from each ballot box (11 bulletins all together), then there is always a candidate whose name is written in each of the 11 chosen bulletins. Prove that there is a ballot box, in which all the bulletins contain the name of the same candidate.
The cells of a \(15 \times 15\) square table are painted red, blue and green. Prove that there are two lines which at least have the same number of cells of one colour.
In a chess tournament, each participant played two games with each of the other participants: one with white pieces, the other with black. At the end of the tournament, it turned out that all of the participants scored the same number of points (1 point for a victory, \(\frac{1}{2}\) a point for a draw and 0 points for a loss). Prove that there are two participants who have won the same number of games using white pieces.
In a mathematical olympiad, \(m>1\) candidates solved \(n>1\) problems. Each candidate solved a different number of problems to all the others. Each problem was solved by a different number of candidates to all the others. Prove that one of the candidates solved exactly one problem.
Is it possible for the mean of some 35 whole numbers to equal \(6.35\)?
What is the largest number of counters that can be put on the cells of a chessboard so that on each horizontal, vertical and diagonal (not only on the main ones) there is an even number of counters?
At all rational points of the real line, integers are arranged. Prove that there is a segment such that the sum of the numbers at its ends does not exceed twice the number on its middle.
100 queens, that cannot capture each other, are placed on a \(100 \times 100\) chessboard. Prove that at least one queen is in each \(50 \times 50\) corner square.
An airline flew exactly 10 flights each day over the course of 92 days. Each day, each plane flew no more than one flight. It is known that for any two days in this period there will be exactly one plane which flew on both those days. Prove that there is a plane that flew every day in this period.