Školjka

%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.

%V0 Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity $$f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2$$ Proposed by Japan

%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 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 $\mathbb{R}^+$ 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 Let $n$ be an even positive integer. Prove that there exists a positive integer $k$ such that $$k = f(x) \cdot (x+1)^n + g(x) \cdot (x^n + 1)$$ for some polynomials $f(x), g(x)$ having integer coefficients. If $k_0$ denotes the least such $k,$ determine $k_0$ as a function of $n,$ i.e. show that $k_0 = 2^q$ where $q$ is the odd integer determined by $n = q \cdot 2^r, r \in \mathbb{N}.$ Note: This is variant A6' of the three variants given for this problem.