ILL 1987 Problem 6

Let f be a function that satisfies the following conditions:

$(i)$ If $x > y$ and $f(y) - y \geq v \geq f(x) - x$, then $f(z) = v + z$, for some number $z$ between $x$ and $y$. $(ii)$ The equation $f(x) = 0$ has at least one solution, and among the solutions of this equation, there is one that is not smaller than all the other solutions; $(iii)$ $f(0) = 1$. $(iv)$ $f(1987) \leq 1988$. $(v)$ $f(x)f(y) = f(xf(y) + yf(x) - xy)$.

Find $f(1987)$.

Proposed by Australia.