5. Each pair of numbers $x$ and $y$ is assigned a number $x * y$. Find $1993 * 1935$, given that for any three numbers $x, y$ and $z$ the identities $x * x=0$ and $x *(y * z)=(x * y)+z$ are satisfied.
5. Let's take $y=z$ in the second identity. Then we get

$$
(x * y)+y=x *(y * y)=x * 0
$$

Thus, $x * y=x * 0 - y$. It remains to compute $x * 0$. For this, take $x=y=z$ in the second identity:

$$
x * 0=x *(x * x)=x * x + x=0 + x=x.
$$

Thus, $x * y=x * 0 - y=x - y$. Therefore, $1993 * 1935=1993 - 1935=58$.

Comment. Check that if you take $x * y=x - y$, then both identities are indeed satisfied.