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 We consider the division of a chess board $8 \times 8$ in p disjoint rectangles which satisfy the conditions: a) every rectangle is formed from a number of full squares (not partial) from the 64 and the number of white squares is equal to the number of black squares. b) the numbers $\ a_{1}, \ldots, a_{p}$ of white squares from $p$ rectangles satisfy $a_1, , \ldots, a_p.$ Find the greatest value of $p$ for which there exists such a division and then for that value of $p,$ all the sequences $a_{1}, \ldots, a_{p}$ for which we can have such a division. Moderator says: see http://www.artofproblemsolving.com/Foru ... 41t=58591

%V0 We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

%V0 For each finite set $U$ of nonzero vectors in the plane we define $l(U)$ to be the length of the vector that is the sum of all vectors in $U.$ Given a finite set $V$ of nonzero vectors in the plane, a subset $B$ of $V$ is said to be maximal if $l(B)$ is greater than or equal to $l(A)$ for each nonempty subset $A$ of $V.$ (a) Construct sets of 4 and 5 vectors that have 8 and 10 maximal subsets respectively. (b) Show that, for any set $V$ consisting of $n \geq 1$ vectors the number of maximal subsets is less than or equal to $2n.$

%V0 Let $x_1$, $x_2$, $\ldots$, $x_n$ be real numbers satisfying the conditions: $ |x_1 + x_2 + \dots + x_n| = 1 $ and $|x_i| \leq \frac{n+1}{2}$, for $i = 1, 2, \dots, n$ Show that there exists a permutation $y_1$, $y_2$, $\ldots$, $y_n$ of $x_1$, $x_2$, $\ldots$, $x_n$ such that $$| y_1 + 2 y_2 + \cdots + n y_n | \leq \frac {n + 1}{2}.$$

%V0 We colour every square of the $2009 \times 2009$ board with one of $n$ colours (we do not have to use every colour). A colour is called connected if either there is only one square of that colour or any two squares of the colour can be reached from one another by a sequence of moves of a chess queen without intermediate stops at squares having another colour (a chess quen moves horizontally, vertically or diagonally). Find the maximum $n$, such that for every colouring of the board at least on colour present at the board is connected.