Problems

Can there exist two functions $f$ and $g$ that take only integer values such that for any integer $x$ the following relations hold:

a) $f(f(x))=x$, $g(g(x))=x$, $f(g(x))>x$, $g(f(x))>x$?

b) $f(f(x))<x$, $g(g(x))<x$, $f(g(x))>x$, $g(f(x))>x$?

The functions $f$ and $g$ are defined on the entire number line and are reciprocal. It is known that $f$ is represented as a sum of a linear and a periodic function: $f(x)=kx+h(x)$, where $k$ is a number, and $h$ is a periodic function. Prove that $g$ is also represented in this form.

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)?

The function $f(x)$ is defined for all real numbers, and for any $x$ the equalities $f(x+2)=f(2-x)$ and $f(x+7)=f(7-x)$ are satisfied. Prove that $f(x)$ is a periodic function.

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