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.
A board of size \(2005\times2005\) is divided into square cells with a side length of 1 unit. Some board cells are numbered in some order by numbers 1, 2, ... so that from any non-numbered cell there is a numbered cell within a distance of less than 10. Prove that there can be found two cells with a distance between them of less than 150, which are numbered by numbers that differ by more than 23. (The distance between the cells is the distance between their centres.)
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.
On an island there are 1,234 residents, each of whom is either a knight (who always tells the truth) or a liar (who always lies). One day, all of the inhabitants of the island were broken up into pairs, and each one said: “He is a knight!" or “He is a liar!" about his partner. Could it eventually turn out to be that the number of “He is a knight!" and “He is a liar!" phrases is the same?
Solve the inequality: \(|x + 2000| <|x - 2001|\).
Determine all natural numbers \(m\) and \(n\) such as \(m! + 12 = n^2\).
Solving the problem: “What is the solution of the expression \(x^{2000} + x^{1999} + x^{1998} + 1000x^{1000} + 1000x^{999} + 1000x^{998} + 2000x^3 + 2000x^2 + 2000x + 3000\) (\(x\) is a real number) if \(x^2 + x + 1 = 0\)?”, Vasya got the answer of 3000. Is Vasya right?
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.
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.
If you have a piece of paper and some scissors, is it possible to cut a hole out of the paper that’s big enough for you to crawl through?