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

Suppose that in each issue of our journal in the “Quantum” problem book there are five mathematics problems. We denote by $f(x,y)$ the number of the first of the problems of the $x$-th issue for the $y$-th year. Write a general formula for $f(x,y)$, where $1\ge x\ge 12$ and $1970\ge y\ge 1989$. Solve the equation $f(x,y)=y$. For example, $f(6,1970)=26$. Since $1989$, the number of tasks has become less predictable. For example, in recent years, half the issues have 5 tasks, and in other issues there are 10. Even the number of magazine issues has changed, no longer being 12 but now 6.

Author: V.A. Popov

On the interval $[0;1]$ a function $f$ is given. This function is non-negative at all points, $f(1)=1$ and, finally, for any two non-negative numbers $x_1$ and $x_2$ whose sum does not exceed 1, the quantity $f(x_1+x_2)$ does not exceed the sum of $f(x_1)$ and $f(x_2)$.

a) Prove that for any number $x$ on the interval $[0;1]$, the inequality $f(x_2)\le 2x$ holds.

b) Prove that for any number $x$ on the interval $[0;1]$, the $f(x_2)\le 1.9x$ must be true?

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.

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)?

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 $f(-1)$.