Problems

The function $f$ is such that for any positive $x$ and $y$ the equality $f(xy)=f(x)+f(y)$ holds. Find $f(2007)$ if $f(1/2007)=1$.

For what natural numbers $a$ and $b$ is the number ${\log }_{a}b$ rational?

Prove that for every natural number $n>1$ the equality: $$\lfloor n^{1/2}\rfloor +\lfloor n^{1/3}\rfloor +\cdots +\lfloor n^{1/n}\rfloor =\lfloor {\log }_{2}n\rfloor +\lfloor {\log }_{3}n\rfloor +\cdots +\lfloor {\log }_{n}n\rfloor$$ is satisfied.

The function $f(x)$ on the interval $[a,b]$ is equal to the maximum of several functions of the form $y=C\times {10}^{-|x-d|}$ (where $d$ and $C$ are different, and all $C$ are positive). It is given that $f(a)=f(b)$. Prove that the sum of the lengths of the sections on which the function increases is equal to the sum of the lengths of the sections on which the function decreases.