Problems

To each pair of numbers $x$ and $y$ some number $x\ast y$ is placed in correspondence. Find $1993\ast 1935$ if it is known that for any three numbers $x,y,z$, the following identities hold: $x\ast x=0$ and $x\ast (y\ast z)=(x\ast y)+z$.

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 if the irreducible rational fraction $p/q$ is a root of the polynomial $P(x)$ with integer coefficients, then $P(x)=(qx-p)Q(x)$, where the polynomial $Q(x)$ also has integer coefficients.

One of the roots of the equation $x^2+ax+b=0$ is $1+\sqrt{3}$. Find $a$ and $b$ if you know that they are rational.

Prove that if $(p,q)=1$ and $p/q$ is a rational root of the polynomial $P(x)=a_n{x}^n+\cdots +a_{1}x+a_0$ with integer coefficients, then

a) $a_0$ is divisible by $p$;

b) $a_n$ is divisible by $q$.

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?

Author: A. Khrabrov

Do there exist integers $a$ and $b$ such that

a) the equation $x^2+ax+b=0$ does not have roots, and the equation $\lfloor x^2\rfloor +ax+b=0$ does have roots?

b) the equation $x^2+2ax+b=0$ does not have roots, and the equation $\lfloor x^2\rfloor +2ax+b=0$ does have roots?

Note that here, square brackets represent integers and curly brackets represent non-integer values or 0.

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 $x_1\ge x_3\ge x_2$.

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.

Find $x^3+y^3$ if $x+y=5$ and $x+y+x^{2}y+x{y}^2=24$.