Problems

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.

For what natural numbers $a$ and $b$ is the number ${\log }_{a}b$ rational?

Prove that if $(m,10)=1$, then there is a repeated unit $E_n$ that is divisible by $m$. Will there be infinitely many repeated units?

Prove the following formulae are true: $$\begin{array}{rl}{a}^{n+1}-b^{n+1}& =(a-b)(a^n+a^{n-1}b+\cdots +b^n);\\ a^{2n+1}+b^{2n+1}& =(a+b)(a^{2n}-a^{2n-1}b+a^{2n-2}{b}^2-\cdots +b^{2n}).\end{array}$$

Derive from the theorem in question 61013 that $\sqrt{17}$ is an irrational number.

For a given polynomial $P(x)$ we describe a method that allows us to construct a polynomial $R(x)$ that has the same roots as $P(x)$, but all multiplicities of 1. Set $Q(x)=(P(x),P^{\prime }(x))$ and $R(x)=P(x)Q^{-1}(x)$. Prove that

a) all the roots of the polynomial $P(x)$ are the roots of $R(x)$;

b) the polynomial $R(x)$ has no multiple roots.

Construct the polynomial $R(x)$ from the problem 61019 if:

a) $P(x)=x^6-6x^4-4x^3+9x^2+12x+4$;

b)$P(x)=x^5+x^4-2x^3-2x^2+x+1$.

Let it be known that all the roots of some equation $x^3+p{x}^2+qx+r=0$ are positive. What additional condition must be satisfied by its coefficients $p,q$ and $r$ in order for it to be possible to form a triangle from segments whose lengths are equal to these roots?