Školjka

%V0 A tetrahedron is called a MEMO-tetrahedron if all six sidelengths are different positive integers where one of them is $2$ and one of them is $3$. Let $l(T)$ be the sum of the sidelengths of the tetrahedron $T$. (a) Find all positive integers $n$ so that there exists a MEMO-Tetrahedron $T$ with $l(T)=n$. (b) How many pairwise non-congruent MEMO-tetrahedrons $T$ satisfying $l(T)=2007$ exist? Two tetrahedrons are said to be non-congruent if one cannot be obtained from the other by a composition of reflections in planes, translations and rotations. (It is not neccessary to prove that the tetrahedrons are not degenerate, i.e. that they have a positive volume).

%V0 Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. Proposed by David Monk, United Kingdom

%V0 For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition. Proposed by Gerhard Woeginger, Netherlands

%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 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 A circle $S$ bisects a circle $S'$ if it cuts $S'$ at opposite ends of a diameter. $S_A$, $S_B$,$S_C$ are circles with distinct centers $A, B, C$ (respectively). Show that $A, B, C$ are collinear iff there is no unique circle $S$ which bisects each of $S_A$, $S_B$,$S_C$ . Show that if there is more than one circle $S$ which bisects each of $S_A$, $S_B$,$S_C$ , then all such circles pass through two fixed points. Find these points. Original Statement: A circle $S$ is said to cut a circle $\Sigma$ diametrically if and only if their common chord is a diameter of $\Sigma.$ Let $S_A, S_B, S_C$ be three circles with distinct centres $A,B,C$ respectively. Prove that $A,B,C$ are collinear if and only if there is no unique circle $S$ which cuts each of $S_A, S_B, S_C$ diametrically. Prove further that if there exists more than one circle $S$ which cuts each $S_A, S_B, S_C$ diametrically, then all such circles $S$ pass through two fixed points. Locate these points in relation to the circles $S_A, S_B, S_C.$