Solve the inequality: \(|x + 2000| <|x - 2001|\).
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.
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?
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?
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.
In the town of Ely, all the families have separate houses. On one fine day, each family moved into another, one of the houses house that used to be occupied by other families. They afterwards decided to paint all houses in red, blue or green colors in such a way that for each family the colour of the new and old houses would not match. Is this always possible to paint te houses in such a way, regardless of how families decided to move?