Problems

The numerical function $f$ is such that for any $x$ and $y$ the equality $f(x+y)=f(x)+f(y)+80xy$ holds. Find $f(1)$ if $f(0.25)=2$.

Does there exist a function $f(x)$ defined for all real numbers such that $f(\sin x)+f(\cos x)=\sin x$?

Are there functions $p(x)$ and $q(x)$ such that $p(x)$ is an even function and $p(q(x))$ is an odd function (different from identically zero)?

For all real $x$ and $y$, the equality $f(x^2+y)=f(x)+f(y^2)$ holds. Find $f(-1)$.

The function $f(x)$ is defined on the positive real $x$ and takes only positive values. It is known that $f(1)+f(2)=10$ and $f(a+b)=f(a)+f(b)+2\sqrt{f(a)f(b)}$ for any $a$ and $b$. Find $f(2^{2011})$.