Školjka

%V0 A number of $n$ rectangles are drawn in the plane. Each rectangle has parallel sides and the sides of distinct rectangles lie on distinct lines. The rectangles divide the plane into a number of regions. For each region $R$ let $v(R)$ be the number of vertices. Take the sum $\sum v(R)$ over the regions which have one or more vertices of the rectangles in their boundary. Show that this sum is less than $40n$.

%V0 A cake has the form of an $n$ x $n$ square composed of $n^{2}$ unit squares. Strawberries lie on some of the unit squares so that each row or column contains exactly one strawberry; call this arrangement $A$. Let $B$ be another such arrangement. Suppose that every grid rectangle with one vertex at the top left corner of the cake contains no fewer strawberries of arrangement $B$ than of arrangement $A$. Prove that arrangement $B$ can be obtained from $A$ by performing a number of switches, defined as follows: A switch consists in selecting a grid rectangle with only two strawberries, situated at its top right corner and bottom left corner, and moving these two strawberries to the other two corners of that rectangle.

%V0 A rectangle $D$ is partitioned in several ($\ge2$) rectangles with sides parallel to those of $D$. Given that any line parallel to one of the sides of $D$, and having common points with the interior of $D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $D$'s boundary. Author: unknown author, Japan

%V0 In the Cartesian coordinate plane define the strips $S_n = \{(x,y)|n\le x < n + 1\}$, $n\in\mathbb{Z}$ and color each strip black or white. Prove that any rectangle which is not a square can be placed in the plane so that its vertices have the same color. IMO Shortlist 2007 Problem C5 as it appears in the official booklet:In the Cartesian coordinate plane define the strips $S_n = \{(x,y)|n\le x < n + 1\}$ for every integer $n.$ Assume each strip $S_n$ is colored either red or blue, and let $a$ and $b$ be two distinct positive integers. Prove that there exists a rectangle with side length $a$ and $b$ such that its vertices have the same color. Edited by Orlando Döhring Author: Radu Gologan and Dan Schwarz, Romania

%V0 In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a box. Two boxes intersect if they have a common point in their interior or on their boundary. Find the largest $n$ for which there exist $n$ boxes $B_1$, $\ldots$, $B_n$ such that $B_i$ and $B_j$ intersect if and only if $i\not\equiv j\pm 1\pmod n$. Proposed by Gerhard Woeginger, Netherlands

%V0 On a $999\times 999$ board a limp rook can move in the following way: From any square it can move to any of its adjacent squares, i.e. a square having a common side with it, and every move must be a turn, i.e. the directions of any two consecutive moves must be perpendicular. A non-intersecting route of the limp rook consists of a sequence of pairwise different squares that the limp rook can visit in that order by an admissible sequence of moves. Such a non-intersecting route is called cyclic, if the limp rook can, after reaching the last square of the route, move directly to the first square of the route and start over. How many squares does the longest possible cyclic, non-intersecting route of a limp rook visit? Proposed by Nikolay Beluhov, Bulgaria