IMO Shortlist 1987 problem 1

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Dodao/la: arhiva
2. travnja 2012.
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.
Izvor: Međunarodna matematička olimpijada, shortlist 1987