Prove that amongst the numbers of the form \[19991999\dots 19990\dots 0\] – that is 1999 a number of times, followed by a number of 0s – there will be at least one divisible by 2001.
There are three piles of rocks: in the first pile there are 10 rocks, 15 in the second pile and 20 in the third pile. In this game (with two players), in one turn a player is allowed to divide one of the piles into two smaller piles. The loser is the one who cannot make a move. Which player would be the winner?
In the first pile there are 100 sweets and in the second there are 200. Consider the game with two players where: in one turn a player can take any amount of sweets from one of the piles. The winner is the one who takes the last sweet. Which player would win by using the correct strategy?
In the dense dark forest ten sources of dead water are erupting from the ground: named from #1 to #10. Of the first nine sources, dead water can be taken by everyone, but the source #10 is in the cave of the dark wizard, from which no one, except for the dark wizard himself, can collect water. The taste and color of dead water is no different from ordinary water, however, if a person drinks from one of the sources, then he will die. Only one thing can save him: if he then drinks poison from a source whose number is greater. For example, if he drinks from the seventh source, then he must necessarily drink poison from the #8, #9 or #10 sources. If he doesn’t drink poison from the seventh source, but does from the ninth, only the poison from the source #10 will save him. And if he originally drinks the tenth poison, then nothing will help him now. Robin Hood summoned the dark wizard to a duel. The terms of the duel were as follows: each brings with him a mug of liquid and gives it to his opponent. The dark wizard was delighted: “Hurray, I will give him poison #10, and Robin Hood can not be saved! And I’ll drink the poison, which Robin Hood brings to me, then ill drink the #10 poison and that will save me!” On the appointed day, both opponents met at the agreed place. They honestly exchanged mugs and drank what was in them. However, afterwards erupted the joy and surprise of the inhabitants of the dark forest, when it turned out that the dark wizard had died, and Robin Hood remained alive! Only the Wise Owl was able to guess how Robin Hood had managed to defeat dark wizard. Try and guess as well.
Is it possible to cut out such a hole in a sheet of paper through which a person could climb through?
The best student in the class, Katie, made up a huge number, writing out in a row all of the natural numbers from 1 to 500: \[123 \dots 10111213 \dots 499500.\] The second-best student, Tom, erased the first 500 digits of this number. What do you think, what number does the remaining number begin with?
Fred and George together with their mother were decorating the Christmas tree. So that they would not fight, their mother gave each brother the same number of decorations and branches. Fred tried to hang one decoration on each branch, but he needed one more branch for his last decoration. George tried to hang two toys on each branch, but one branch was empty. What do you think, how many branches and how many decorations did the mother give to her sons?
The farmer must transport across a river a wolf, a goat and a cabbage. The boat accommodates one person, and with him/her either a wolf, a goat, or a cabbage. If you leave the goat and the wolf unattended, the wolf will eat the goat. If you leave cabbage and goat without supervision, the goat will eat the cabbage. How can the farmer transport his cargo across the river?
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.
There is a 12-litre barrel filled with beer, and two empty kegs of 5 and 8 litres. Try using these kegs to:
a) divide the beer into two parts of 3 and 9 litres;
b) divide the beer into two equal parts.