Problems

Two different numbers $x$ and $y$ (not necessarily integers) are such that $x^2-2000x=y^2-2000y$. Find the sum of $x$ and $y$.

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 $x\ast (y\ast z)=(x\ast y)+z$.

Prove that for any natural number $a_1>1$ there exists an increasing sequence of natural numbers $a_1,a_2,a_3,\dots$, for which $a_1^2+a_2^2+\cdots +a_k^2$ is divisible by $a_1+a_2+\cdots +a_k$ for all $k\ge 1$.

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.