Aladdin visited all of the points on the equator, moving to the east, then to the west, and sometimes instantly moving to the diametrically opposite point on Earth. Prove that there was a period of time during which the difference in distances traversed by Aladdin to the east and to the west was not less than half the length of the equator.
The numbers \(a\) and \(b\) are such that the first equation of the system \[\begin{aligned} \sin x + a & = bx \\ \cos x &= b \end{aligned}\] has exactly two solutions. Prove that the system has at least one solution.
The numbers \(a\) and \(b\) are such that the first equation of the system \[\begin{aligned} \cos x &= ax + b \\ \sin x + a &= 0 \end{aligned}\] has exactly two solutions. Prove that the system has at least one solution.
Determine all natural numbers \(m\) and \(n\) such as \(m! + 12 = n^2\).
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.
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 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?
At the cat show, 10 male cats and 19 female cats sit in a row where next to each female cat sits a fatter male cat. Prove that next to each male cat is a female cat, which is thinner than it.