Let $ABC$ be an acute angled triangle and $CD$ be the altitude through $C$. If $AB = 8$ and $CD = 6$, find the distance between the midpoints of $AD$ and $BC$.

Using $ AB$ and $ AC$ as diameters, two semi-circles are constructed respectively outside the acute triangle $ ABC$. $ AH \perp BC$ at $ H$, $ D$ is any point on side $ BC$ ($ D$ is not coinside with $ B$ or $ C$ ), through $ D$, construct $ DE \parallel AC$ and $ DF \parallel AB$ with $ E$ and $ F$ on the two semi-circles respectively. Show that $ D$, $ E$, $ F$ and $ H$ are concyclic.

Inside the triangle $ ABC $ a point $ M $ is given. The line $ BM $ meets the side $ AC $ at $ N $. The point $ K $ is symmetrical to $ M $ with respect to $ AC $. The line $ BK $ meets $ AC $ at $ P $. If $ \angle AMP = \angle CMN $, prove that $ \angle ABP=\angle CBN $.

Let $ABC$ be a triangle and $A_1, B_1, C_1$ points on $BC, CA, AB$, respectively, such that the lines $AA_1, BB_1, CC_1$ meet at a single point. It is known that $A, B_1, A_1, B$ are concyclic and $B, C_1, B_1, C$ are concyclic. Prove that a) $C, A_1, C_1, A$ are concyclic, b) $AA_1,, BB_1, CC_1$ are the heights of $ABC$.

Two circles $\omega , \Omega$ intersect in distinct points $A,B$ a line through $B$ intersects $\omega , \Omega$ in $C,D$ respectively such that $B$ lies between $C,D$ another line through $B$ intersects $\omega , \Omega$ in $E,F$ respectively such that $E$ lies between $B,F$ and $FE=CD$. Furthermore $CF$ intersects $\omega , \Omega$ in $P,Q$ respectively and $M,N$ are midpoints of the arcs $PB,QB$. Prove that $CNMF$ is a cyclic quadrilateral.

Let $ ABC $ be a triangle. The perpendicular in $ B $ of the bisector of the angle $ \angle ABC $ intersects the bisector of the angle $ \angle BAC $ in $ M. $ Show that $ MC $ is perpendicular to the bisector of $ \angle BCA. $

Alice and Bob live on the edges and vertices of the unit cube. Alice begins at point $(0, 0, 0)$ and Bob begins at $(1, 1, 1)$. Every second, each of them chooses one of the three adjacent corners and walks at a constant rate of $1$ unit per second along the edge until they reach the corner, after which they repeat the process. What is the expected amount of time in seconds before Alice and Bob meet?

A non self-intersecting polygon is given in a Cartesian coordinate system such that its perimeter contains no lattice points, and its vertices have no integer coordinates. A point is called semi-integer if exactly one of its coordinates is an integer. Let $P_1, P_2,\ldots, P_k$ denote the semi-integer points on the perimeter of the polygon. Let ni denote the floor of the non-integer coordinate of $P_i$. Prove that integers $n_1,n_2,\ldots ,n_k$ can be divided into two groups with the same sum. [i]Proposed by Áron Bán-Szabó, Budapest[/i]

Let $ABC$ be a triangle, and let $E$ and $F$ be two arbitrary points on the sides $AB$ and $AC$, respectively. The circumcircle of triangle $AEF$ meets the circumcircle of triangle $ABC$ again at point $M$. Let $D$ be the reflection of point $M$ across the line $EF$ and let $O$ be the circumcenter of triangle $ABC$. Prove that $D$ is on $BC$ if and only if $O$ belongs to the circumcircle of triangle $AEF$.

A circle with center $O$ is tangent to the sides of the angle with the vertex $A$ at the points B and C. Let M be a point on the larger of the two arcs $BC$ of this circle (different from $B$ and $C$) such that $M$ does not lie on the line $AO$. Lines $BM$ and $CM$ intersect the line $AO$ at the points $P$ and $Q$ respectively. Let $K$ be the foot of the perpendicular drawn from $P$ to $AC$ and $L$ be the foot of the perpendicular drawn from $Q$ to $AB$. Prove that the lines $OM$ and $KL$ are perpendicular.

Find all values of positive integers $a$ and $b$ such that it is possible to put $a$ ones and $b$ zeros in every of vertices in polygon with $a+b$ sides so it is possible to rotate numbers in those vertices with respect to primary position and after rotation one neighboring $0$ and $1$ switch places and in every other vertices other than those two numbers remain the same.

Two walkers are at the same altitude in a range of mountains. The path joining them is piecewise linear with all its vertices above the two walkers. Can they each walk along the path until they have changed places, so that at all times their altitudes are equal?

Let $ABCD$ be a cyclic quadrilateral. Assume that the points $Q, A, B, P$ are collinear in this order, in such a way that the line $AC$ is tangent to the circle $ADQ$, and the line $BD$ is tangent to the circle $BCP$. Let $M$ and $N$ be the midpoints of segments $BC$ and $AD$, respectively. Prove that the following three lines are concurrent: line $CD$, the tangent of circle $ANQ$ at point $A$, and the tangent to circle $BMP$ at point $B$.

Find the greatest integer $n$, such that there are $n+4$ points $A$, $B$, $C$, $D$, $X_1,\dots,~X_n$ in the plane with $AB\ne CD$ that satisfy the following condition: for each $i=1,2,\dots,n$ triangles $ABX_i$ and $CDX_i$ are equal.

For a nonnegative integer $n$, define $a_n$ to be the positive integer with decimal representation \[1\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}1\mbox{.}\] Prove that $\frac{a_n}{3}$ is always the sum of two positive perfect cubes but never the sum of two perfect squares. [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 7)[/i]

Given an acute non-isosceles triangle $ABC$ with circumcircle $\Gamma$. $M$ is the midpoint of segment $BC$ and $N$ is the midpoint of arc $BC$ of $\Gamma$ (the one that doesn't contain $A$). $X$ and $Y$ are points on $\Gamma$ such that $BX\parallel CY\parallel AM$. Assume there exists point $Z$ on segment $BC$ such that circumcircle of triangle $XYZ$ is tangent to $BC$. Let $\omega$ be the circumcircle of triangle $ZMN$. Line $AM$ meets $\omega$ for the second time at $P$. Let $K$ be a point on $\omega$ such that $KN\parallel AM$, $\omega_b$ be a circle that passes through $B$, $X$ and tangents to $BC$ and $\omega_c$ be a circle that passes through $C$, $Y$ and tangents to $BC$. Prove that circle with center $K$ and radius $KP$ is tangent to 3 circles $\omega_b$, $\omega_c$ and $\Gamma$. [i]Proposed by Tran Quan - Vietnam[/i]

Let $XY$ be a chord of a circle $\Omega$, with center $O$, which is not a diameter. Let $P, Q$ be two distinct points inside the segment $XY$, where $Q$ lies between $P$ and $X$. Let $\ell$ the perpendicular line drawn from $P$ to the diameter which passes through $Q$. Let $M$ be the intersection point of $\ell$ and $\Omega$, which is closer to $P$. Prove that $$ MP \cdot XY \ge 2 \cdot QX \cdot PY$$

Let $\vartriangle ABC$ be an acute triangle with altitudes $AA_1, BB_1, CC_1$ and orthocenter $H$. Let $K, L$ be the midpoints of $BC_1, CB_1$. Let $\ell_A$ be the external angle bisector of $\angle BAC$. Let $\ell_B, \ell_C$ be the lines through $B, C$ perpendicular to $\ell_A$. Let $\ell_H$ be the line through $H$ parallel to $\ell_A$. Prove that the centers of the circumcircles of $\vartriangle A_1B_1C_1, \vartriangle AKL$ and the rectangle formed by $\ell_A, \ell_B, \ell_C, \ell_H$ lie on the same line.

The rectangle with vertices $(0,0)$, $(0,3)$, $(2,0)$ and $(2,3)$ is rotated clockwise through a right angle about the point $(2,0)$, then about $(5,0)$, then about $(7,0$), and finally about $(10,0)$. The net effect is to translate it a distance $10$ along the $x$-axis. The point initially at $(1,1)$ traces out a curve. Find the area under this curve (in other words, the area of the region bounded by the curve, the $x$-axis and the lines parallel to the $y$-axis through $(1,0)$ and $(11,0)$).

Let $ABC$ be an acute-angled triangle with circumcenter $O$. The circle $S$ through $A,B$, and $O$ intersects $AC$ and $BC$ again at points $P$ and $Q$ respectively. Prove that $CO \perp PQ$.

In the quadrilateral $ABCD$, $AB = BC$ . The point $E$ lies on the line $AB$ is such that $BD= BE$ and $AD \perp DE$. Prove that the perpendicular bisectors to segments $AD, CD$ and $CE$ intersect at one point.

Three convex polygons are given on a plane. Prove that there is no line cutting all the polygons if and only if each of the polygons can be separated from the other two by a line.

Three pairwise externally tangent circles $\omega_1,\omega_2$ and $\omega_3$ are given. Let $K_{12}$ be the point of tangency between $\omega_1$ and $\omega_2$ and define $K_{23}$ and $K_{31}$ similarly. Consider the point $A_1$ on $\omega_1$. Let $A_2$ be the second intersection of the line $A_1K_{12}$ with $\omega_2$. The line $A_2K_{23}$ then intersects $\omega_3$ the second time at $A_3$, and then line $A_3K_{31}$ intersects $\omega_1$ again at $A_4$ and so on. [list=a] [*]Prove that after six steps, the process will loop; that is, $A_7=A_1$. [*]Prove that the lines $A_1A_2$ and $A_4A_5$ are perpendicular. [*]Prove that the triples of lines $A_1A_2,A_3A_4$ and $A_5A_6$ and $A_2A_3,A_4A_5$ and $A_6A_1$ intersect at two diametrically opposite points on the circle $(K_{12}K_{23}K_{31})$. [/list] [i]Proposed by E. Morozov[/i]

In order for Mateen to walk a kilometer ($1000$m) in his rectangular backyard, he must walk the length $25$ times or walk its perimeter $10$ times. What is the area of Mateen's backyard in square meters? $\text{(A)}\ 40 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 400 \qquad \text{(D)}\ 500 \qquad \text{(E)}\ 1000$

Let $ \triangle{ABC}$ be a triangle with $ AB\not\equal{}AC$. The incircle with centre $ I$ touches $ BC$, $ CA$, $ AB$ at $ D$, $ E$, $ F$, respectively. Furthermore let $ M$ the midpoint of $ EF$ and $ AD$ intersect the incircle at $ P\not\equal{}D$. Show that $ PMID$ ist cyclic.