Školjka

%V0 Show that for any finite set $S$ of distinct positive integers, we can find a set $T$ ⊇ $S$ such that every member of $T$ divides the sum of all the members of $T$. Original Statement: A finite set of (distinct) positive integers is called a DS-set if each of the integers divides the sum of them all. Prove that every finite set of positive integers is a subset of some DS-set.

%V0 Ten points are marked in the plane so that no three of them lie on a line. Each pair of points is connected with a segment. Each of these segments is painted with one of $k$ colors, in such a way that for any $k$ of the ten points, there are $k$ segments each joining two of them and no two being painted with the same color. Determine all integers $k$, $1\leq k\leq 10$, for which this is possible.

%V0 Suppose that every integer has been given one of the colours red, blue, green or yellow. Let $x$ and $y$ be odd integers so that $|x| \neq |y|$. Show that there are two integers of the same colour whose difference has one of the following values: $x,y,x+y$ or $x-y$.

%V0 Let $p$ and $q$ be relatively prime positive integers. A subset $S$ of $\{0, 1, 2, \ldots \}$ is called ideal if $0 \in S$ and for each element $n \in S,$ the integers $n + p$ and $n + q$ belong to $S.$ Determine the number of ideal subsets of $\{0, 1, 2, \ldots \}.$

%V0 Given $n$ real numbers $x_1$, $x_2$, ..., $x_n$, and $n$ further real numbers $y_1$, $y_2$, ..., $y_n$. The entries $a_{ij}$ (with $1\leq i,\;j\leq n$) of an $n\times n$ matrix $A$ are defined as follows: $a_{ij}=\left\{\begin{array}{c}1\text{\ \ \ \ \ \ if\ \ \ \ \ \ }x_{i}+y_{j}\geq 0;\\ 0\text{\ \ \ \ \ \ if\ \ \ \ \ \ }x_{i}+y_{j}<0.\end{array}\right.$ Further, let $B$ be an $n\times n$ matrix whose elements are numbers from the set $\left\{0;\ 1\right\}$ satisfying the following condition: The sum of all elements of each row of $B$ equals the sum of all elements of the corresponding row of $A$; the sum of all elements of each column of $B$ equals the sum of all elements of the corresponding column of $A$. Show that in this case, $A = B$. comment (This one is from the ISL 2003, but in any case, the official problems and solutions - in German - are already online, hence I take the liberty to post it here.) Darij

%V0 Let $S = \{x_1, x_2, \ldots, x_{k + l}\}$ be a $(k + l)$-element set of real numbers contained in the interval $[0, 1]$; $k$ and $l$ are positive integers. A $k$-element subset $A\subset S$ is called nice if $$\left |\frac {1}{k}\sum_{x_i\in A} x_i - \frac {1}{l}\sum_{x_j\in S\setminus A} x_j\right |\le \frac {k + l}{2kl}$$ Prove that the number of nice subsets is at least $\dfrac{2}{k + l}\dbinom{k + l}{k}$. Proposed by Andrey Badzyan, Russia