Let $ ABCD $ be a sqare and $ E $ be a point situated on the segment $ BD, $ but not on the mid. Denote by $ H $ and $ K $ the orthocenters of $ ABE, $ respectively, $ ADE. $ Show that $ \overrightarrow{BH}=\overrightarrow{KD} . $

On a $4 \times 4 \times 3$ rectangular parallelepiped, vertices $A$, $B$, and $C$ are adjacent to vertex $D$. The perpendicular distance from $D$ to the plane containing $A$, $B$, and $C$ is closest to $\text{(A)}\ 1.6 \qquad \text{(B)}\ 1.9 \qquad \text{(C)}\ 2.1 \qquad \text{(D)}\ 2.7 \qquad \text{(E)}\ 2.9$

We define a [i]pseudo-inverse[/i] $B\in \mathcal M_n(\mathbb C)$ of a matrix $A\in\mathcal M_n(\mathbb C)$ a matrix which fulfills the relations \[ A = ABA \quad \text{ and } \quad B=BAB. \] a) Prove that any square matrix has at least a pseudo-inverse. b) For which matrix $A$ is the pseudo-inverse unique? [i]Marius Cavachi[/i]

In an acute triangle $\triangle{ABC}$, let $AE$ and $BF$ be highs of it, and $H$ its orthocenter. The symmetric line of $AE$ with respect to the angle bisector of $\sphericalangle{A}$ and the symmetric line of $BF$ with respect to the angle bisector of $\sphericalangle{B}$ intersect each other on the point $O$. The lines $AE$ and $AO$ intersect again the circuncircle to $\triangle{ABC}$ on the points $M$ and $N$ respectively. Let $P$ be the intersection of $BC$ with $HN$; $R$ the intersection of $BC$ with $OM$; and $S$ the intersection of $HR$ with $OP$. Show that $AHSO$ is a paralelogram.

For an integer $n\geq 3$, let $\theta=2\pi/n$. Evaluate the determinant of the $n\times n$ matrix $I+A$, where $I$ is the $n\times n$ identity matrix and $A=(a_{jk})$ has entries $a_{jk}=\cos(j\theta+k\theta)$ for all $j,k$.

Let $S$ be a set of positive integers, each of them having exactly $100$ digits in base $10$ representation. An element of $S$ is called [i]atom[/i] if it is not divisible by the sum of any two (not necessarily distinct) elements of $S$. If $S$ contains at most $10$ atoms, at most how many elements can $S$ have?

8.8, 9.8, 11.8 a) 99 boxes contain apples and oranges. Prove that we can choose 50 boxes in such a way that they contain at least half of all apples and half of all oranges. b) 100 boxes contain apples and oranges. Prove that we can choose 34 boxes in such a way that they contain at least a third of all apples and a third of all oranges. c) 100 boxes contain apples, oranges and bananas. Prove that we can choose 51 boxes in such a way that they contain at least half of all apples, and half of all oranges and half of all bananas. ([i]I. Bogdanov, G. Chelnokov, E. Kulikov[/i])

We define an operation $\oplus$ on the set $\{0, 1\}$ by \[ 0 \oplus 0 = 0 \,, 0 \oplus 1 = 1 \,, 1 \oplus 0 = 1 \,, 1 \oplus 1 = 0 \,.\] For two natural numbers $a$ and $b$, which are written in base $2$ as $a = (a_1a_2 \ldots a_k)_2$ and $b = (b_1b_2 \ldots b_k)_2$ (possibly with leading 0's), we define $a \oplus b = c$ where $c$ written in base $2$ is $(c_1c_2 \ldots c_k)_2$ with $c_i = a_i \oplus b_i$, for $1 \le i \le k$. For example, we have $7 \oplus 3 = 4$ since $ 7 = (111)_2$ and $3 = (011)_2$. For a natural number $n$, let $f(n) = n \oplus \left[ n/2 \right]$, where $\left[ x \right]$ denotes the largest integer less than or equal to $x$. Prove that $f$ is a bijection on the set of natural numbers.

Let $P$ and $P^{\prime }$ be two parallelograms with equal area, and let their sidelengths be $a,$ $b$ and $a^{\prime },$ $b^{\prime }.$ Assume that $a^{\prime }\leq a\leq b\leq b^{\prime },$ and moreover, it is possible to place the segment $b^{\prime }$ such that it completely lies in the interior of the parallelogram $P.$ Show that the parallelogram $P$ can be partitioned into four polygons such that these four polygons can be composed again to form the parallelogram $% P^{\prime }.$

$AB$ is a chord of length $6$ of a circle centred at $O$ and of radius $5$. Let $PQRS$ denote the square inscribed in the sector $OAB$ such that $P$ is on the radius $OA$, $S$ is on the radius $OB$ and $Q$ and $R$ are points on the arc of the circle between $A$ and $B$. Find the area of $PQRS$.

Let $ABCD$ be a convex quadrilateral.Let diagonals $AC$ and $BD$ intersect at $P$. Let $PE,PF,PG$ and $PH$ are altitudes from $P$ on the side $AB,BC,CD$ and $DA$ respectively. Show that $ABCD$ has a incircle if and only if $\frac{1}{PE}+\frac{1}{PG}=\frac{1}{PF}+\frac{1}{PH}.$

$ABC$ is a triangle with $\angle{B}>45^{\circ}$ , $\angle{C}>45^{\circ}$. We draw the isosceles triangles $CAM,BAN$ on the sides $AC,AB$ and outside the triangle, respectively, such that $\angle{CAM}=\angle{BAN}=90^{\circ}$. And we draw isosceles triangle $BPC$ on the side $BC$ and inside the triangle such that $\angle{BPC}=90^{\circ}$. Prove that $\Delta{MPN}$ is an isosceles triangle, too, and $\angle{MPN}=90^{\circ}$.

A lamp is situated at point $A$ and shines inside the cube. A (massive) square is hung on the midpoints of the 4 vertical faces. What's the area of its shadow? [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=285[/img]

Determine whether or not there exist two different sets $A,B$, each consisting of at most $2011^2$ positive integers, such that every $x$ with $0 < x < 1$ satisfies the following inequality: \[\left| \sum_{a \in A} x^a - \sum_{b \in B} x^b \right| < (1-x)^{2011}.\]