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 $f(1/2007)=1$.

Are there any irrational numbers $a$ and $b$ such that the degree of $a^b$ is a rational number?

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

Let’s call a natural number good if in its decimal record we have the numbers 1, 9, 7, 3 in succession, and bad if otherwise. (For example, the number 197,639,917 is bad and the number 116,519,732 is good.) Prove that there exists a positive integer $n$ such that among all $n$-digit numbers (from ${10}^{n-1}$ to ${10}^{n-1}$) there are more good than bad numbers.

Try to find the smallest possible $n$.

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.