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.

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.

A two-player game with matches. There are 37 matches on the table. In each turn, a player is allowed to take no more than 5 matches. The winner of the game is the player who takes the final match. Which player wins, if the right strategy is used?

Two weighings. There are 7 coins which are identical on the surface, including 5 real ones (all of the same weight) and 2 counterfeit coins (both of the same weight, but lighter than the real ones). How can you find the 3 real coins with the help of two weighings on scales without weights?

Multiplication of numbers. Restore the following example of the multiplication of natural numbers if it is known that the sum of the digits of both factors is the same.

Restore the example of the multiplication.

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.