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 $f(1/2007)=1$.

On a function $f(x)$, defined on the entire real line, it is known that for any $a>1$ the function $f(x)+f(ax)$ is continuous on the whole line. Prove that $f(x)$ is also continuous on the whole line.

Find all functions $f(x)$ defined for all real values of $x$ and satisfying the equation $2f(x)+f(1-x)=x^2$.

Suppose that there are 15 prime numbers forming an arithmetic progression with a difference of $d$. Prove that $d>30,000$.