Problems

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.

Let $m$ and $n$ be integers. Prove that $mn(m+n)$ is an even number.

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 $b$.

The numbers $1,2,\dots ,99$ are written on 99 cards. Then the cards are shuffled and placed with the number facing down. On the blank side of the cards, the numbers $1,2,\dots ,99$ are once again written.

The sum of the two numbers on each card are calculated, and the product of these 99 summations is worked out. Prove that the end result will be an even number.

Prove that any $n$ numbers $x_1,\dots ,x_n$ that are not pairwise congruent modulo $n$, represent a complete system of residues, modulo $n$.

Prove that for any natural number there is a multiple of it, the decimal notation of which consists of only 0 and 1.

Without calculating the answer to $2^{30}$, prove that it contains at least two identical digits.

Write the following rational numbers in the form of decimal fractions: a) $\frac{1}{7}$; b) $\frac{2}{7}$; c) $\frac{1}{14}$; d) $\frac{1}{17}$.

Let the number $\alpha$ be given by the decimal:

a) $0.101001000100001000001\dots$;

b) $0.123456789101112131415\dots$.

Will this number be rational?