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.

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

Compute the following: $$\frac{(2001\times 2021+100)(1991\times 2031+400)}{{2011}^4}.$$

Prove that, if $b=a-1$, then $$(a+b)(a^2+b^2)(a^4+b^4)\cdots (a^{32}+b^{32})=a^{64}-b^{64}.$$

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

The number $x$ is such a number that exactly one of the four numbers $a=x-\sqrt{2}$, $b=x-1/x$, $c=x+1/x$, $d=x^2+2\sqrt{2}$ is not an integer. Find all such $x$.

The numbers $x$, $y$ and $z$ are such that all three numbers $x+yz$, $y+zx$ and $z+xy$ are rational, and $x^2+y^2=1$. Prove that the number $xy{z}^2$ is also rational.

Solve the inequality: $\lfloor x\rfloor \times \{x\}<x-1$.