Školjka

%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 $i = 0, \ldots, n - 2.$

%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 Let $b, m, n$ be positive integers such that $b > 1$ and $m \neq n.$ Prove that if $b^m - 1$ and $b^n - 1$ have the same prime divisors, then $b + 1$ is a power of 2.

%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 $p$ be a prime number and $f$ an integer polynomial of degree $d$ such that $f(0) = 0,f(1) = 1$ and $f(n)$ is congruent to $0$ or $1$ modulo $p$ for every integer $n$. Prove that $d\geq p - 1$.

%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.