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 $n$ and $k$ be positive integers such that $\frac{1}{2} n < k \leq \frac{2}{3} n.$ Find the least number $m$ for which it is possible to place $m$ pawns on $m$ squares of an $n \times n$ chessboard so that no column or row contains a block of $k$ adjacent unoccupied squares.

%V0 Let $f(k)$ be the number of all non-negative integers $n$ satisfying the following conditions: (1) The integer $n$ has exactly $k$ digits in the decimal representation (where the first digit is not necessarily non-zero!), i. e. we have $0 \leq n <10^k$. (2) These $k$ digits of n can be permuted in such a way that the resulting number is divisible by 11. Show that for any positive integer number $m,$ we have $f\left(2m\right) = 10 f\left(2m - 1\right)$.

%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