Školjka

%V0 A solitaire game is played on an $m\times n$ rectangular board, using $mn$ markers which are white on one side and black on the other. Initially, each square of the board contains a marker with its white side up, except for one corner square, which contains a marker with its black side up. In each move, one may take away one marker with its black side up, but must then turn over all markers which are in squares having an edge in common with the square of the removed marker. Determine all pairs $(m,n)$ of positive integers such that all markers can be removed from the board.

%V0 Let $p >3$ be a prime number. For each nonempty subset $T$ of $\{0,1,2,3, \ldots , p-1\}$, let $E(T)$ be the set of all $(p-1)$-tuples $(x_1, \ldots ,x_{p-1} )$, where each $x_i \in T$ and $x_1+2x_2+ \ldots + (p-1)x_{p-1}$ is divisible by $p$ and let $|E(T)|$ denote the number of elements in $E(T)$. Prove that $$|E(\{0,1,3\})| \geq |E(\{0,1,2\})|$$ with equality if and only if $p = 5$.

%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 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 $\alpha < \frac {3 - \sqrt {5}}{2}$ be a positive real number. Prove that there exist positive integers $n$ and $p > \alpha \cdot 2^n$ for which one can select $2 \cdot p$ pairwise distinct subsets $S_1, \ldots, S_p, T_1, \ldots, T_p$ of the set $\{1,2, \ldots, n\}$ such that $S_i \cap T_j \neq \emptyset$ for all $1 \leq i,j \leq p$ Author: Gerhard Wöginger, Austria

%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