Školjka

%V0 Seventeen people correspond by mail with one another-each one with all the rest. In their letters only three different topics are discussed. each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.

%V0 Prove that we can find an infinite set of positive integers of the from $2^n-3$ (where $n$ is a positive integer) every pair of which are relatively prime.

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