Is it possible to cut out such a hole in a sheet of paper through which a person could climb through?
Prove that the following facts are true for any graph:
a) The sum of degrees of all vertices is equal to twice the number of edges (and therefore it is even);
b) The number of vertices of odd degree is even.
During a chess tournament, some of the players played an odd number of games. Prove that the number of such players is even.
48 blacksmiths must shoe 60 horses. Each blacksmith spends 5 minutes on one horseshoe. What is the shortest time they should spend on the work? (Note that a horse can not stand on two legs.)
a) Prove that within any 6 whole numbers there will be two that have a difference between them that is a multiple of 5.
b) Will this statement remain true if instead of the difference we considered the total?
Is the number \(10^{2002} + 8\) divisible by 9?
Is the sum of the numbers \(1 + 2 + 3 + \dots + 1999\) divisible by 1999?
A schoolboy told his friend Bob:
“We have thirty-five people in the class. And imagine, each of them is friends with exactly eleven classmates...”
“It cannot be,” Bob, the winner of the mathematical Olympiad, answered immediately. Why did he decide this?
In an ordinary set of dominoes, there are 28 tiles. How many tiles would a set of dominoes contain if the values indicated on the tiles did not range from 0 to 6, but from 0 to 12?
A class contains 38 pupils. Prove that within the class there will be at least 4 pupils born in the same month.