Sixty children came to the maths circle, coming from several different schools, and each school sent at least two children. Show that if among any group of ten children, there are always at least three who attend the same school, then among all sixty children there must be at least fifteen who come from one particular school.
The people of Wonderland are having an election. Each voter writes the names of 10 candidates on their ballot. No name can be written twice on the same ballot.
There are 11 ballot boxes in total and each box has at least one ballot inside. The March Hare, who is counting the votes, notices something:
If he takes one ballot from each box (so 11 altogether), there is always at least one candidate whose name appears on all 11 of those papers.
Prove that there is at least one ballot box and one candidate such that every ballot in that box contains the name of that 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.