Hong Kong Preliminary Selection Contest

From $0=f(f(0))=f(b)=ab+b=(a+1)b$, we get $a=-1$ or $b=0$. If $b=0$, i.e. $f(x)=ax$, then we have $9=f(f(f(4)))=f(f(4a))=f(4a^2)=4a^3$, which has no solution as $a$ is an integer.

So we must have $a=-1$. Then $f(x)=-x+b$, and hence $f(f(x))=-(-x+b)+b=x$. It follows that $f(f(f(x)))=f(f(x))=x$ for all $x$, and thus the answer is $$1+2+\cdots +2014=\frac{2014\times 2015}{2}=2029105.$$