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

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.