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 $n$.

The parliament of a certain country has two houses with an equal number of members. In order to make a decision on an important issue all the members voted and there were no abstentions. When the chairman announced that the decision had been taken with a 23-vote advantage, the opposition leader declared that the results had been rigged. How did he know it?

Is it possible to find natural numbers $x$, $y$ and $z$ which satisfy the equation $28x+30y+31z=365$?

Is there a sequence of natural numbers in which every natural number occurs exactly once, and for any $k=1,2,3,\dots$ the sum of the first $k$ terms of the sequence is divisible by $k$?

Given a board (divided into squares) of the size: a) $10\times 12$, b) $9\times 10$, c) $9\times 11$, consider the game with two players where: in one turn a player is allowed to cross out any row or any column if there is at least one square not crossed out. The loser is the one who cannot make a move. Is there a winning strategy for one of the players?

Prove that for a real positive $\alpha$ and a positive integer $d$, $\lfloor \alpha /d\rfloor =\lfloor \lfloor \alpha \rfloor /d\rfloor$ is always satisfied.

Prove that if $p$ is a prime number and $1\le k\le p-1$, then $(\genfrac{}{}{0ex}{}{p}{k})$ is divisible by $p$.

Prove that if $p$ is a prime number, then $(a+b{)}^p-a^p-b^p$ is divisible by $p$ for any integers $a$ and $b$.

Prove that if $(p,q)=1$ and $p/q$ is a rational root of the polynomial $P(x)=a_n{x}^n+\cdots +a_{1}x+a_0$ with integer coefficients, then

a) $a_0$ is divisible by $p$;

b) $a_n$ is divisible by $q$.