Problems

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 $d>30,000$.

Method of iterations. In order to approximately solve an equation, it is allowed to write $f(x)=x$, by using the iteration method. First, some number $x_0$ is chosen, and then the sequence $\{x_n\}$ is constructed according to the rule $x_{n+1}=f(x_n)$ ($n\ge 0$). Prove that if this sequence has the limit $x\ast =\underset{n\to \mathrm{\infty }}{lim}{x}_n$, and the function $f(x)$ is continuous, then this limit is the root of the original equation: $f(x^{\ast })=x^{\ast }$.

Find the largest value of the expression $a+b+c+d-ab-bc-cd-da$, if each of the numbers $a$, $b$, $c$ and $d$ belongs to the interval $[0,1]$.

We consider a function $y=f(x)$ defined on the whole set of real numbers and satisfying $f(x+k)\times (1-f(x))=1+f(x)$ for some number $k\ne 0$. Prove that $f(x)$ is a periodic function.

$f(x)$ is an increasing function defined on the interval $[0,1]$. It is known that the range of its values belongs to the interval $[0,1]$. Prove that, for any natural $N$, the graph of the function can be covered by $N$ rectangles whose sides are parallel to the coordinate axes so that the area of each is $1/N^2$. (In a rectangle we include its interior points and the points of its boundary).

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.