Školjka

%V0 A function $f : I \to \mathbb R$, defined on an interval $I$, is called concave if $f(\theta x + (1 - \theta)y) \geq \theta f(x) + (1 - \theta)f(y)$ for all $x, y \in I$ and $0 \leq \theta \leq 1$. Assume that the functions $f_1, \ldots , f_n$, having all nonnegative values, are concave. Prove that the function $(f_1f_2 \cdots f_n)^{1/n}$ is concave.

%V0 Prove that the functional equations $$f(x + y) = f(x) + f(y),$$ $$\text{and} \qquad f(x + y + xy) = f(x) + f(y) + f(xy) \quad (x, y \in \mathbb R)$$ are equivalent.

%V0 Let f be a function that satisfies the following conditions: $(i)$ If $x > y$ and $f(y) - y \geq v \geq f(x) - x$, then $f(z) = v + z$, for some number $z$ between $x$ and $y$. $(ii)$ The equation $f(x) = 0$ has at least one solution, and among the solutions of this equation, there is one that is not smaller than all the other solutions; $(iii)$ $f(0) = 1$. $(iv)$ $f(1987) \leq 1988$. $(v)$ $f(x)f(y) = f(xf(y) + yf(x) - xy)$. Find $f(1987)$. Proposed by Australia.

%V0 Let $a, b \in \mathbb{N}$ with $1 \leq a \leq b,$ and $M = \left[\frac {a + b}{2} \right].$ Define a function $f: \mathbb{Z} \mapsto \mathbb{Z}$ by $$f(n) = \begin{cases} n + a, & \text{if } n \leq M, \\ n - b, & \text{if } n \geq M. \end{cases}$$ Let $f^1(n) = f(n),$ $f_{i + 1}(n) = f(f^i(n)),$ $i = 1, 2, \ldots$ Find the smallest natural number $k$ such that $f^k(0) = 0.$

%V0 Let $\mathbb{R}^+$ be the set of all non-negative real numbers. Given two positive real numbers $a$ and $b,$ suppose that a mapping $f: \mathbb{R}^+ \mapsto \mathbb{R}^+$ satisfies the functional equation: $$f(f(x)) + af(x) = b(a + b)x.$$ Prove that there exists a unique solution of this equation.

%V0 Does there exist functions $f,g: \mathbb{R}\to\mathbb{R}$ such that $f(g(x)) = x^2$ and $g(f(x)) = x^k$ for all real numbers $x$ a) if $k = 3$? b) if $k = 4$?