Problems

A board of size $2005\times 2005$ 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.)

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: $|x+2000|<|x-2001|$.

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.

Is it possible to cut out such a hole in a sheet of paper through which a person could climb 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 $1+2+3+\cdots +1999$ 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?