Problems

Find all functions $f(x)$ defined for all positive $x$, taking positive values and satisfying the equality $f(x^y)=f(x{)}^f(y)$ for any positive $x$ and $y$.

The function $f(x)$ is defined and satisfies the relationship $(x-1)f((x=1)/(x-1))-f(x)=x$ for all $x\ne 1$. Find all such functions.

Find all the functions $f:\mathbb{R}\to \mathbb{R}$ which satisfy the inequality $f(x+y)+f(y+z)+f(z+x)\ge 3f(x+2y+3z)$ for all $x,y,z$.

Is there a bounded function $f:\mathbb{R}\to \mathbb{R}$ such that $f(1)>0$ and $f(x)$ satisfies the inequality $f^2(x+y)\ge f^2(x)+2f(xy)+f^2(y)$ for all $x,y\in \mathbb{R}$?

For which $\alpha$ does there exist a function $f:\mathbb{R}\to \mathbb{R}$ that is not a constant, such that $f(\alpha (x+y))=f(x)+f(y)$?

On a function $f(x)$ defined on the whole line of real numbers, 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.

Does there exist a function $f(x)$ defined for all $x\in \mathbb{R}$ and for all $x,y\in \mathbb{R}$ satisfying the inequality $|f(x+y)+\sin x+\sin y|<2$?

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.

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

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 $f(1/2007)=1$.