China Mathematical Competition

Question

It is given that f(x) is a function defined on R, satisfying f(1) = 1, and for any x R, f(x+5) f(x)+5, and f(x+1) f(x)+1. If g(x) = f(x)+1-x, then g(2002) = 2cm.

Accepted Answer

We determine f(2002) first. From the conditions given, we have aligned f(x)+5 & f(x+5) f(x+4)+1 & f(x+3)+2 f(x+2)+3 & f(x+1)+4 f(x)+5. aligned Thus the equality holds for all. So we have f(x+1)=f(x)+1. Hence, from f(1)=1, we get f(2)=2, f(3)=3, ..., f(2002)=2002. Therefore, g(2002)=f(2002)+1-2002=1.