Školjka

%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 Let $P(x)$ be a polynomial with real coefficients such that $P(x) > 0$ for all $x \geq 0.$ Prove that there exists a positive integer n such that $(1 + x)^n \cdot P(x)$ is a polynomial with nonnegative coefficients.

%V0 In town $A,$ there are $n$ girls and $n$ boys, and each girl knows each boy. In town $B,$ there are $n$ girls $g_1, g_2, \ldots, g_n$ and $2n - 1$ boys $b_1, b_2, \ldots, b_{2n-1}.$ The girl $g_i,$ $i = 1, 2, \ldots, n,$ knows the boys $b_1, b_2, \ldots, b_{2i-1},$ and no others. For all $r = 1, 2, \ldots, n,$ denote by $A(r),B(r)$ the number of different ways in which $r$ girls from town $A,$ respectively town $B,$ can dance with $r$ boys from their own town, forming $r$ pairs, each girl with a boy she knows. Prove that $A(r) = B(r)$ for each $r = 1, 2, \ldots, n.$

%V0 Let $a_1\geq \cdots \geq a_n \geq a_{n + 1} = 0$ be real numbers. Show that $$\sqrt {\sum_{k = 1}^n a_k} \leq \sum_{k = 1}^n \sqrt k (\sqrt {a_k} - \sqrt {a_{k + 1}}).$$ Proposed by Romania

%V0 For every integer $n \geq 2$ determine the minimum value that the sum $\sum^n_{i=0} a_i$ can take for nonnegative numbers $a_0, a_1, \ldots, a_n$ satisfying the condition $a_0 = 1,$ $a_i \leq a_{i+1} + a_{i+2}$ for $i = 0, \ldots, n - 2.$