A circle center $O$ is inscribed in the quadrilateral $ABCD$. $AB$ is parallel to and longer than $CD$ and has midpoint $M$. The line $OM$ meets $CD$ at $F$. $CD$ touches the circle at $E$. Show that $DE = CF$ iff $AB = 2CD$.

Let $\triangle ABC$ be an acute angle triangle and $\omega$ be its circumcircle. Let $N$ be a point on arc $AC$ not containing $B$ and $S$ be a point on line $AB$. The line tangent to $\omega$ at $N$ intersects $BC$ at $T$, $NS$ intersects $\omega$ at $K$. Assume that $\angle NTC = \angle KSB$. Prove that $CK\parallel AN \parallel TS$.

A circle $k$ is given with a non-diameter chord $AC$. On the tangent at point $A$ select point $X \ne A$ and mark $D$ the intersection of the circle $k$ with the interior of the line $XC$ (if any). Let $B$ a point in circle $k$ such that quadrilateral $ABCD$ is a trapezoid . Determine the set of intersections of lines $BC$ and $AD$ belonging to all such trapezoids.

Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.) [asy]import three; import math; size(180); defaultpen(linewidth(.8pt)); currentprojection=orthographic(2,0.2,1); triple A=(0,0,1); triple B=(sqrt(2)/2,sqrt(2)/2,0); triple C=(sqrt(2)/2,-sqrt(2)/2,0); triple D=(-sqrt(2)/2,-sqrt(2)/2,0); triple E=(-sqrt(2)/2,sqrt(2)/2,0); triple F=(0,0,-1); draw(A--B--E--cycle); draw(A--C--D--cycle); draw(F--C--B--cycle); draw(F--D--E--cycle,dotted+linewidth(0.7));[/asy]$ \textbf{(A)}\ 210 \qquad \textbf{(B)}\ 560 \qquad \textbf{(C)}\ 840 \qquad \textbf{(D)}\ 1260 \qquad \textbf{(E)}\ 1680$

The points $A_{1},A_{2},B_{1},B_{2},C_{1},C_{2}$ are on the sides $AB$, $BC$ and $AC$ of an acute triangle $ABC$ such that $AA_{1} = A_{1}A_{2} = A_{2}B = \frac{1}{3} AB$, $BB_{1} = B_{1}B_{2} = B_{2}C = \frac{1}{3}BC$ and $CC_{1} = C_{1}C_{2} = C_{2}A = \frac{1}{3} AC$. Let $k_{A}, k_{B}$ and $k_{C}$ be the circumcircles of the triangles $AA_{1}C_{2}$, $BB_{1}A_{2}$ and $CC_{1}B_{2}$ respectively. Furthermore, let $a_{B}$ and $a_{C}$ be the tangents to $k_{A}$ at $A_{1}$ and $C_{2}$, $b_{C}$ and $b_{A}$ the tangents to $k_{B}$ at $B_{1}$ and $A_{2}$ and $c_{A}$ and $c_{B}$ the tangents to $k_{C}$ at $C_{1}$ and $B_{2}$. Show that the perpendicular lines from the intersection points of $a_{B}$ and $b_{A}$, $b_{C}$ and $c_{B}$, $c_{A}$ and $a_{C}$ to $AB$, $BC$ and $CA$ respectively are concurrent.

On the sides of the triangle $ABC$, three similar isosceles triangles $ABP \ (AP = PB)$, $AQC \ (AQ = QC)$, and $BRC \ (BR = RC)$ are constructed. The first two are constructed externally to the triangle $ABC$, but the third is placed in the same half-plane determined by the line $BC$ as the triangle $ABC$. Prove that $APRQ$ is a parallelogram.

Let $ABC$ be a triangle, the point $E$ is in the segment $AC$, the point $F$ is in the segment $AB$ and $P=BE\cap CF$. Let $D$ be a point such that $AEDF$ is a parallelogram, Prove that $D$ is in the side $BC$, if and only if, the triangle $BPC$ and the quadrilateral $AEPF$ have the same area.

Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

Let $ABC$ be a scalene triangle, and $A_o$, $B_o,$ $C_o$ be the midpoints of $BC$, $CA$, $AB$ respectively. The bisector of angle $C$ meets $A_oCo$ and $B_oC_o$ at points $B_1$ and $A_1$ respectively. Prove that the lines $AB_1$, $BA_1$ and $A_oB_o$ concur.

Side $AB$ of triangle $ABC$ was divided into $n$ equal parts (dividing points $B_0 = A, B_1, B_2, ..., B_n = B$), and side $AC$ of this triangle was divided into $(n + 1)$ equal parts (dividing points $C_0 = A, C_1, C_2, ..., C_{n+1} = C$). Colored are the triangles $C_iB_iC_{i+1}$ (where $i = 1,2, ..., n$). What part of the area of the triangle is painted over?

The incircle of the triangle $ABC$ touches the sides $AC$ and $BC$ at points $K$ and $L$, respectively. the $B$-excircle touches the side $AC$ of this triangle at point $P$. The segment $AL$ intersects the inscribed circle for the second time at point $S$. Line $KL$ intersects the circumscribed circle of triangle $ASK$ for the second at point $M$. Prove that $PL = PM$.

Let $n$ and $c$ be coprime positive integers. For any integer $i$, denote by $i' $ the remainder of division of product $ci$ by $n$. Let $A_o.A_1,A_2,...,A_{n-1}$ be a regular $n$-gon. Prove that a) if $A_iA_j \parallel A_kA_i$ then $A_{i'}A_{j'} \parallel A_{k'}A_{i'}$ b) if $A_iA_j \perp A_kA_l$ then $A_{i'}A_{j'} \perp A_{k'}A_{l'}$

Four points in space $A,B,C,D$ satisfy that $|AB|=3,|BC|=7,|CD|=11,|DA|=9$, then the number of values of $\overrightarrow{AC}\cdot\overrightarrow{BD}$ is $\text{(A)}$ Only one. $\text{(B)}$ Two. $\text{(C)}$ Three. $\text{(D)}$ Infinitely many.

Phillip is trying to make a two-dimensional donut, but in a fun way: He is trying to make a donut shaped in a way that the inner circle of the donut is inscribed inside a pentagon, and the outer circle of the donut circumscribes the same pentagon. This pentagon has a side length of $6$. The area of Phillip's donut is of the form $a \pi$. Find $a$. (Note that $\sin 54^o= \frac{\sqrt5+1}{4}$ )

Given $7$ points inside or on a regular hexagon, show that three of them form a triangle with area $\le 1/6$ the area of the hexagon.

Let $M$ be the midpoint of the arc $BAC$ of the circumcircle of the triangle $ABC,$ $I$ the incenter of the triangle $ABC$ and $L$ a point on the side $BC$ such that $AL$ is bisector. The line $MI$ cuts the circumcircle again at $K.$ The circumcircle of the triangle $AKL$ cuts the line $BC$ again at $P.$ Prove that $\angle AIP = 90^{\circ}.$

Holding a rectangular sheet of paper $ABCD$, Prair folds triangle $ABD$ over diagonal $BD$, so that the new location of point $A$ is $A'$. She notices that $A'C =\frac13 BD$. If the area of $ABCD$ is $27\sqrt2$, find $BD$.

a) Two legs of an angle $\alpha$ on a plane are mirrors. Prove that after several reflections in the mirrors any ray leaves in the direction opposite the one from which it came if and only if $\alpha = \frac{90^o}{n}$ for an integer $n$. Find the number of reflections. b) Given three planar mirrors in space forming an octant (trihedral angle with right planar angles), prove that any ray of light coming into this mirrored octant leaves it, after several reflections in the mirrors, in the direction opposite to the one from which it came. Find the number of reflections.

Let $A, B, C$ and $D$ be points on the circle $\omega$ such that $ABCD$ is a convex quadrilateral. Suppose that $AB$ and $CD$ intersect at a point $E$ such that $A$ is between $B$ and $E$ and that $BD$ and $AC$ intersect at a point $F$. Let $X \ne D$ be the point on $\omega$ such that $DX$ and $EF$ are parallel. Let $Y$ be the reflection of $D$ through $EF$ and suppose that $Y$ is inside the circle $\omega$. Show that $A, X$, and $Y$ are collinear.

Given a triangle $ABC$. $D$ is a moving point on the edge $BC$. Point $E$ and Point $F$ are on the edge $AB$ and $AC$, respectively, such that $BE=CD$ and $CF=BD$. The circumcircle of $\triangle BDE$ and $\triangle CDF$ intersects at another point $P$ other than $D$. Prove that there exists a fixed point $Q$, such that the length of $QP$ is constant.

Let $D$ be the point on the base $BC$ of an isosceles $\vartriangle ABC$ triangle such that $\frac{\left| BD \right|}{\left| DC \right|}=\text{ }2$, and let $P$ be the point on the segment $\left[ AD \right]$ such that $\angle BAC=\angle BPD$. Prove that $\angle DPC=\frac{1}{2}\angle BAC$.

A rectangular field has dimensions $120$ meters and $192$ meters. You want to divide it into equal square plots. The measure of the sides of these squares must be an integer number . In addition, you want to place a post in each corner of plot. Determine the smallest number of plots in which you can divide the land and the number of posts needed. [hide=Original wording]Un terreno de forma rectangular de 120 metros por 192 metros se quiere dividir en parcelas cuadradas iguales sin que sobre terreno. La medida de los lados de estos cuadrados debe ser un nu´mero entero. Adem´as se desea colocar un poste en cada esquina de parcela. Determinar el menor nu´mero de parcelas en que se puede dividir el terreno y el nu´mero de postes que se necesitan.[/hide]

Equal line segments are marked in triangle $ABC$. Find its angles. [img]https://cdn.artofproblemsolving.com/attachments/0/2/bcb756bba15ba57013f1b6c4cbe9cc74171543.png[/img]

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Triangle $ABC$ and straight line $l$ are given at the plane. Construct using a compass and a ruler the straightline which is parallel to $l$ and bisects the area of triangle $ABC.$

Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $\Gamma_4$ be distinct circles such that $\Gamma_1$, $\Gamma_3$ are externally tangent at $P$, and $\Gamma_2$, $\Gamma_4$ are externally tangent at the same point $P$. Suppose that $\Gamma_1$ and $\Gamma_2$; $\Gamma_2$ and $\Gamma_3$; $\Gamma_3$ and $\Gamma_4$; $\Gamma_4$ and $\Gamma_1$ meet at $A$, $B$, $C$, $D$, respectively, and that all these points are different from $P$. Prove that \[ \frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}. \]