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.

Solve the equations in integers:

a) $3x^2+5y^2=345$;

b) $1+x+x^2+x^3=2^y$.

Prove that in a three-digit number, that is divisible by 37, you can always rearrange the numbers so that the new number will also be divisible by 37.

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

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.

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

a) $0.101001000100001000001\dots$;

b) $0.123456789101112131415\dots$.

Will this number be rational?

Prove that in any infinite decimal fraction you can rearrange the numbers so that the resulting fraction becomes a rational number.