Problems

$N$ points are given, no three of which lie on one line. Each two of these points are connected by a segment, and each segment is coloured in one of the $k$ colours. Prove that if $N>\lfloor k!e\rfloor$, then among these points one can choose three such that all sides of the triangle formed by them will be colored in one colour.

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?

There are 4 weights and scales. How many loads that are different by weight can be accurately weighed using these weights, if

a) weights can be placed only on one side of the scales;

b) weights can be placed on both sides of the scales?

Will thought of a number: 1, 2 or 3. You can ask him only one question, to which he can answer “yes”, “no” or “I do not know”. Can you guess the number by asking just one question?

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

Prove that for $n>0$ the polynomial $n{x}^{n+1}-(n+1)x^n+1$ is divisible by $(x-1{)}^2$.

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?