Problems

Prove that there is a number of the form

a) $1989\dots 19890\dots 0$ (the number 1989 is repeated several times, and then there are a few zeros), which is divisible by 1988;

b) $1988\dots 1988$, which is divisible by 1989.

Does there exist a flat quadrilateral in which the tangents of all interior angles are equal?

The numbers $a$ and $b$ are such that the first equation of the system $$\begin{array}{rl}\sin x+a& =bx\\ \cos x& =b\end{array}$$ has exactly two solutions. Prove that the system has at least one solution.

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.)

The numbers $a$ and $b$ are such that the first equation of the system $$\begin{array}{rl}\cos x& =ax+b\\ \sin x+a& =0\end{array}$$ 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.