Školjka

%V0 Let $f$ be a function from the set of real numbers $\mathbb{R}$ into itself such for all $x \in \mathbb{R},$ we have $|f(x)| \leq 1$ and $$f \left( x + \frac{13}{42} \right) + f(x) = f \left( x + \frac{1}{6} \right) + f \left( x + \frac{1}{7} \right).$$ Prove that $f$ is a periodic function (that is, there exists a non-zero real number $c$ such $f(x+c) = f(x)$ for all $x \in \mathbb{R}$).

%V0 Let $\mathbb{R}^+$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^+ \longrightarrow \mathbb{R}^+$ that satisfy the following conditions: - $f(xyz)+f(x)+f(y)+f(z)=f(\sqrt{xy})f(\sqrt{yz})f(\sqrt{zx})$ for all $x,y,z\in\mathbb{R}^+$; - $f(x)<f(y)$ for all $1\le x<y$.

%V0 Find all functions $f\colon\mathbb{R} \rightarrow\mathbb{R}$ satisfying the equation $$f\left(x^2 + y^2 + 2f\left(xy\right)\right) = \left(f\left(x + y\right)\right)^2$$ for all $x,y\in \mathbb{R}$.

%V0 Let $f: \mathbb{R}\to\mathbb{N}$ be a function which satisfies $f\left(x + \dfrac{1}{f(y)}\right) = f\left(y + \dfrac{1}{f(x)}\right)$ for all $x$, $y\in\mathbb{R}$. Prove that there is a positive integer which is not a value of $f$. Proposed by Žymantas Darbėnas (Zymantas Darbenas), Lithania

%V0 Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that $$f\left(x-f(y)\right)>yf(x)+x$$ Proposed by Igor Voronovich, Belarus

%V0 Find all functions $f: \mathbb{R} \to \mathbb{R}$, such that $$f(xf(y)) + f(f(x) + f(y)) = yf(x) + f(x + f(y))$$ holds for all $x$, $y \in \mathbb{R}$, where $\mathbb{R}$ denotes the set of real numbers.