$6$ open disks in the plane are such that the center of no disk lies inside another. Show that no point lies inside all $6$ disks.

A very well known family of mathematicians has three children called [i]Antonia, Bernhard[/i] and [i]Christian[/i]. Each evening one of the children has to do the dishes. One day, their dad decided to construct of plan that says which child has to do the dishes at which day for the following $55$ days. Let $x$ be the number of possible such plans in which Antonia has to do the dishes on three consecutive days at least once. Furthermore, let $y$ be the number of such plans in which there are three consecutive days in which Antonia does the dishes on the first, Bernhard on the second and Christian on the third day. Determine, whether $x$ and $y$ are different and if so, then decide which of those is larger.

In an acute-angled triangle \(ABC\) \((AC > BC)\) with altitude \(AD\), the following points are marked: \(H\) - the orthocenter, \(O\) - the circumcenter, \(K\) - the midpoint of side \(AB\). Inside the triangle \(\triangle ADC\), there is a point \(P\) such that the following equality holds: \[ \angle KPD + \angle ACB = 2 \angle OPH = 180^{\circ} \] Prove that \[ BH = 2PD \] [i]Proposed by Vadym Solomka[/i]

Find all real numbers $a$ such that exists function $\mathbb {R} \rightarrow \mathbb {R} $ satisfying the following conditions: 1) $f(f(x)) =xf(x)-ax$ for all real $x$ 2) $f$ is not constant 3) $f$ takes the value $a$

Let $ABC$ be an acute triangle with orthocenter $ H $ and $AB<AC.$ Let $\Omega_1$ be a circle with diameter $AC$ and $\Omega_2$ a circle with diameter $ AB.$ Line $BH$ intersects $\Omega_1$ in points $ D $ and $E$ such that $E$ is not on segment $BH.$ Line $ CH $ intersects $\Omega_2$ in points $ F $ and $G$ such that $G$ is not on segment $CH.$ Prove that the lines $EG, DF$ and $BC$ are concurrent.

[b]2/1/27.[/b] Suppose $a, b,$ and $c$ are distinct positive real numbers such that \begin{align*}abc=1000, \\ bc(1-a)+a(b+c)=110.\end{align*} If $a<1$, show that $10<c<100$.

There are $n$ cities on each side of Hung river, with two-way ferry routes between some pairs of cities across the river. A city is “convenient” if and only if the city has ferry routes to all cities on the other side. The river is “clear” if we can find $n$ different routes so that the end points of all these routes include all $2n$ cities. It is known that Hung river is currently unclear, but if we add any new route, then the river becomes clear. Determine all possible values for the number of convenient cities. [i] Proposed by usjl[/i]

Calculate the sum of the coordinates of all pairs of positive integers $(n, k)$ such that $k\equiv 0, 3\pmod 4$, $n > k$, and $\displaystyle\sum^n_{i = k + 1} i^3 = (96^2\cdot3 - 1)\displaystyle\left(\sum^k_{i = 1} i\right)^2 + 48^2$

Two players play a game on four piles of pebbles labeled with the numbers $1,2,3,4$ respectively. The players take turns in an alternating fashion. On his or her turn, a player selects integers $m$ and $n$ with $1\leq m<n\leq 4$, removes $m$ pebbles from pile $n$, and places one pebble in each of the piles $n-1,n-2,\dots,n-m$. A player loses the game if he or she cannot make a legal move. For each starting position, determine the player with a winning strategy.

Let $T$ be an acute triangle. Inscribe a rectangle $R$ in $T$ with one side along a side of $T.$ Then inscribe a rectangle $S$ in the triangle formed by the side of $R$ opposite the side on the boundary of $T,$ and the other two sides of $T,$ with one side along the side of $R.$ For any polygon $X,$ let $A(X)$ denote the area of $X.$ Find the maximum value, or show that no maximum exists, of $\tfrac{A(R)+A(S)}{A(T)},$ where $T$ ranges over all triangles and $R,S$ over all rectangles as above.

Prove that there are 100 natural number $a_1 < a_2 < ... < a_{99} < a_{100}$ ( $ a_i < 10^6$) such that A , A+A , 2A , A+2A , 2A + 2A are five sets apart ? $A = \{a_1 , a_2 ,... , a_{99} ,a_{100}\}$ $2A = \{2a_i \vert 1\leq i\leq 100\}$ $A+A = \{a_i + a_j \vert 1\leq i<j\leq 100\}$ $A + 2A = \{a_i + 2a_j \vert 1\leq i,j\leq 100\}$ $2A + 2A = \{2a_i + 2a_j \vert 1\leq i<j\leq 100\}$ (20 ponits )

In a simple graph on $n$ vertices, every set of $k$ vertices has an odd number of common neighbours. Prove that $n+k$ must be odd.

Suppose a $m \times m$ square can be divided into $7$ rectangles such that no two rectangles have a common interior point and the side-lengths of the rectangles form the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 \}$. Find the maximum value of $m$.

Let $AA _1$ and $BB_1$ be the altitudes of an isosceles acute-angled triangle $ABC, M$ the midpoint of $AB$. The circles circumscribed around the triangles $AMA_1$ and $BMB_1$ intersect the lines $AC$ and $BC$ at points $K$ and $L$, respectively. Prove that $K, M$, and $L$ lie on the same line.

Let $O$ be the center of the circle circumscribed to $\vartriangle ABC$, and let $ P$ be any point on $BC$ ($P \ne B$ and $P \ne C$). Suppose that the circle circumscribed to $\vartriangle BPO$ intersects $AB$ at $R$ ($R \ne A$ and $R \ne B$) and that the circle circumscribed to $\vartriangle COP$ intersects $CA$ at point $Q$ ($Q \ne C$ and $Q \ne A$). 1) Show that $\vartriangle PQR \sim \vartriangle ABC$ and that$ O$ is orthocenter of $\vartriangle PQR$. 2) Show that the circles circumscribed to the triangles $\vartriangle BPO$, $\vartriangle COP$, and $\vartriangle PQR$ all have the same radius.

Given sequences $a_n=\frac{1}{n}{\sqrt[n] {_{2n}P_n}},\ b_n=\frac{1}{n^2}{\sqrt[n] {_{4n}P_{2n}}}$ and $c_n=\sqrt[n]{\frac{_{8n}P_{4n}}{_{6n}P_{4n}}}$, find $\lim_{n\to\infty} a_n,\ \lim_{n\to\infty} b_n$and $\lim_{n\to\infty} c_n.$

Let $ABCD$ be a cyclic quadrilateral, $O$ be its circumcenter, $P$ be a common points of its diagonals, and $M , N$ be the midpoints of $AB$ and $CD$ respectively. A circle $OPM$ meets for the second time segments $AP$ and $BP$ at points $A_1$ and $B_1$ respectively and a circle $OPN$ meets for the second time segments $CP$ and $DP$ at points $C_1$ and $D_1$ respectively. Prove that the areas of quadrilaterals $AA_1B_1B$ and $CC_1D_1D$ are equal.

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]

In a plane, a square $ABCD$ and a point $P$ located inside it are given. Let a point $ Q$ pass through all sides of the square. Describe the set of all those points $R$ in for which the triangle $PQR$ is equilateral.

The circle inscribed in a triangle $ABC$ touches the sides $BC,CA,AB$ in $D,E, F$, respectively, and $X, Y,Z$ are the midpoints of $EF, FD,DE$, respectively. Prove that the centers of the inscribed circle and of the circles around $XYZ$ and $ABC$ are collinear.

Find all integers $x$ and $y$ such that $2^x+1=y^2$

Solve the equation in positive integers: $2^x-3^y 5^z=1009$.

Let $m\ge2$ be an integer. The sequence $(a_n)_{n\in\mathbb N}$ is defined by $a_0=0$ and $a_n=\left\lfloor\frac nm\right\rfloor+a_{\left\lfloor\frac nm\right\rfloor}$ for all $n$. Determine $\lim_{n\to\infty}\frac{a_n}n$.

A line $g$ and a point $A$ outside $g$ are given in a plane. A point $P$ moves along $g$. Find the locus of the third vertices of equilateral triangles whose two vertices are $A$ and $P$.

Tickets for the football game are $\$10$ for students and $\$15$ for non-students. If $3000$ fans attend and pay $\$36250$, how many students went?