Školjka

%V0 a) Show that the set $\mathbb{Q}^{ + }$ of all positive rationals can be partitioned into three disjoint subsets. $A,B,C$ satisfying the following conditions: $$BA = B; B^2 = C; BC = A;$$ where $HK$ stands for the set $\{hk: h \in H, k \in K\}$ for any two subsets $H, K$ of $\mathbb{Q}^{ + }$ and $H^2$ stands for $HH.$ b) Show that all positive rational cubes are in $A$ for such a partition of $\mathbb{Q}^{ + }.$ c) Find such a partition $\mathbb{Q}^{ + } = A \cup B \cup C$ with the property that for no positive integer $n \leq 34,$ both $n$ and $n + 1$ are in $A,$ that is, $$\text{min} \{n \in \mathbb{N}: n \in A, n + 1 \in A \} > 34.$$

%V0 Let $n,k \in \mathbb{Z}^{+}$ with $k \leq n$ and let $S$ be a set containing $n$ distinct real numbers. Let $T$ be a set of all real numbers of the form $x_1 + x_2 + \ldots + x_k$ where $x_1, x_2, \ldots, x_k$ are distinct elements of $S.$ Prove that $T$ contains at least $k(n-k)+1$ distinct elements.

%V0 Let $n \geq 2, n \in \mathbb{N}$ and $A_0 = (a_{01},a_{02}, \ldots, a_{0n})$ be any $n-$tuple of natural numbers, such that $0 \leq a_{0i} \leq i-1,$ for $i = 1, \ldots, n.$ $n-$tuples $A_1= (a_{11},a_{12}, \ldots, a_{1n}), A_2 = (a_{21},a_{22}, \ldots, a_{2n}), \ldots$ are defined by: $a_{i+1,j} = Card \{a_{i,l}| 1 \leq l \leq j-1, a_{i,l} \geq a_{i,j}\},$ for $i \in \mathbb{N}$ and $j = 1, \ldots, n.$ Prove that there exists $k \in \mathbb{N},$ such that $A_{k+2} = A_{k}.$

%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 Consider $2009$ cards, each having one gold side and one black side, lying on parallel on a long table. Initially all cards show their gold sides. Two player, standing by the same long side of the table, play a game with alternating moves. Each move consists of choosing a block of $50$ consecutive cards, the leftmost of which is showing gold, and turning them all over, so those which showed gold now show black and vice versa. The last player who can make a legal move wins. (a) Does the game necessarily end? (b) Does there exist a winning strategy for the starting player? Proposed by Michael Albert, Richard Guy, New Zealand

%V0 For any integer $n\geq 2$, let $N(n)$ be the maxima number of triples $(a_i, b_i, c_i)$, $i=1, \ldots, N(n)$, consisting of nonnegative integers $a_i$, $b_i$ and $c_i$ such that the following two conditions are satisfied: $a_i+b_i+c_i=n$ for all $i=1, \ldots, N(n)$, If $i\neq j$ then $a_i\neq a_j$, $b_i\neq b_j$ and $c_i\neq c_j$Determine $N(n)$ for all $n\geq 2$. Proposed by Dan Schwarz, Romania