Problems

A numeric set $M$ containing 2003 distinct numbers is such that for every two distinct elements $a,b$ in $M$, the number $a^2+b\sqrt{2}$ is rational. Prove that for any $a$ in $M$ the number $q\sqrt{2}$ is rational.

Prove that the root a of the polynomial $P(x)$ has multiplicity greater than 1 if and only if $P(a)=0$ and $P^{\prime }(a)=0$.

For which $A$ and $B$ does the polynomial $A{x}^{n+1}+B{x}^n+1$ have the number $x=1$ at least two times as its root?

The equations $$\begin{array}{}\text{(1)}& a{x}^2+bx+c=0\end{array}$$ and $$\begin{array}{}\text{(2)}& -a{x}^2+bx+c\end{array}$$ are given. Prove that if $x_1$ and $x_2$ are, respectively, any roots of the equations (1) and (2), then there is a root $x_3$ of the equation $\frac{1}{2}a{x}^2+bx+c$ such that either $x_1\le x_3\le x_2$ or $x_1\ge x_3\ge x_2$.

Prove that if $x_0^4+a_1{x}_0^3+a_2{x}_0^2+a_3{x}_0+a_4$ and $4x_0^3+3a_1{x}_0^2+2a_2{x}_0+a_3=0$ then $x^4+a_1{x}^3+a_2{x}^2+a_{3}x+a_4$ is divisible by $(x-x_0{)}^2$.

Let $n$ numbers are given together with their product $p$. The difference between $p$ and each of these numbers is an odd number.

Prove that all $n$ numbers are irrational.

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?

Find the coefficient of $x$ for the polynomial $(x-a)(x-b)(x-c)\dots (x-z)$.