Problems

There are $n$ integers. Prove that among them either there are several numbers whose sum is divisible by $n$ or there is one number divisible by $n$ itself.

What is the maximum number of pairwise non-parallel segments with endpoints at the vertices of a regular $n$-gon?

Prove that $n^2+1$ is not divisible by $3$ for any natural $n$.

Recall that a natural number $x$ is called prime if $x$ has no divisors except $1$ and itself. Solve the equation with prime numbers $pqr=7(p+q+r)$.

Solve the equation with natural numbers $1+x+x^2+x^3=2y$.

Numbers $a,b,c$ are integers with $a$ and $b$ being coprime. Let us assume that integers $x_0$ and $y_0$ are a solution for the equation $ax+by=c$.

Prove that every solution for this equation has the same form $x=x_0+kb$, $y=y_0-ka$, with $k$ being a random integer.

There are 13 weights. It is known that any 12 of them could be placed in 2 scale cups with 6 weights in each cup in such a way that balance will be held.

Prove the mass of all the weights is the same, if it is known that:

a) the mass of each weight in grams is an integer;

b) the mass of each weight in grams is a rational number;

c) the mass of each weight could be any real (not negative) number.

A monkey escaped from it’s cage in the zoo. Two guards are trying to catch it. The monkey and the guards run along the zoo lanes. There are six straight lanes in the zoo: three long ones form an equilateral triangle and three short ones connect the middles of the triangle sides. Every moment of the time the monkey and the guards can see each other. Will the guards be able to catch the monkey, if it runs three times faster than the guards? (In the beginning of the chase the guards are in one of the triangle vertices and the monkey is in another one.)

Prove that the number of all arrangements of the largest possible amount of peaceful bishops (figures that move on diagonals and don’t threaten each other) on the $8\times 8$ chessboard is an exact square.

Can there exist two functions $f$ and $g$ that take only integer values such that for any integer $x$ the following relations hold:

a) $f(f(x))=x$, $g(g(x))=x$, $f(g(x))>x$, $g(f(x))>x$?

b) $f(f(x))<x$, $g(g(x))<x$, $f(g(x))>x$, $g(f(x))>x$?