In the adjoining plane figure, sides $AF$ and $CD$ are parallel, as are sides $AB$ and $EF$, and sides $BC$ and $ED$. Each side has length of 1. Also, $\measuredangle FAB = \measuredangle BCD = 60^\circ$. The area of the figure is [asy] size(200); defaultpen(linewidth(0.8)); pair A = dir(145), F = A + (0,-1), E = (0,-1), C = dir(35), D = C + (0,-1), B = origin; draw(A--B--C--D--E--F--cycle); label("$A$",A, dir(100)); label("$B$",B,2*N); label("$C$",C,dir(80)); label("$D$",D,dir(0)); label("$E$",E,S); label("$F$",F,W); label("$60^\circ$",A, 6*dir(295)); label("$60^\circ$",C, 6*dir(245)); [/asy] $\displaystyle \textbf{(A)} \ \frac{\sqrt 3}{2} \qquad \textbf{(B)} \ 1 \qquad \textbf{(C)} \ \frac{3}{2} \qquad \textbf{(D)} \ \sqrt{3} \qquad \textbf{(E)} \ 2$

Let $ABC$ be a triangle and $L$, $M$, $N$ be the midpoints of $BC$, $CA$ and $AB$, respectively. The tangent to the circumcircle of $ABC$ at $A$ intersects $LM$ and $LN$ at $P$ and $Q$, respectively. Show that $CP$ is parallel to $BQ$.

Let $ ABC $ be an acutangle triangle and $ D $ any point on the $ BC $ side. Let $ E $ be the symmetrical of $ D $ in $ AC $ and $ F $ is the symmetrical $ D $ relative to $ AB $. $ A $ straight $ ED $ intersects straight $ AB $ at $ G $, while straight $ F D $ intersects the line $ AC $ in $ H $. Prove that the points $ A, E, F, G$ and $ H $ are on the same circumference.

Let $ABCD$ be a rectangle. Circles $C_1$ and $C_2$ are externally tangent to each other. Furthermore, $C_1$ is tangent to $AB$ and $AD$, and $C_2$ is tangent to $CB$ and $CD$. If $AB = 18$ and $BC = 25$, then find the sum of the radii of the circles.

Let $ABC$ be a scalene acute-angled triangle with its incenter $I$ and circumcircle $\Gamma$. Line $AI$ intersects $\Gamma$ for the second time at $M$. Let $N$ be the midpoint of $BC$ and $T$ be the point on $\Gamma$ such that $IN \perp MT$. Finally, let $P $ and $Q$ be the intersection points of $TB $ and $TC$, respectively, with the line perpendicular to $AI$ at $I$. Show that $PB = CQ$. [i]Proposed by Patrik Bak - Slovakia[/i]

$\omega$ is circumcirlce of triangle $ABC$. We draw a line parallel to $BC$ that intersects $AB,AC$ at $E,F$ and intersects $\omega$ at $U,V$. Assume that $M$ is midpoint of $BC$. Let $\omega'$ be circumcircle of $UMV$. We know that $R(ABC)=R(UMV)$. $ME$ and $\omega'$ intersect at $T$, and $FT$ intersects $\omega'$ at $S$. Prove that $EF$ is tangent to circumcircle of $MCS$.

a) Let $ABC$ be a triangle and a point $I$ lies inside the triangle. Assume $\angle IBA > \angle ICA$ and $\angle IBC >\angle ICB$. Prove that, if extensions of $BI$, $CI$ intersect $AC$, $AB$ at $B'$, $C'$ respectively, then $BB' < CC'$. b) Let $ABC$ be a triangle with $AB < AC$ and angle bisector $AD$. Prove that for every point $I, J$ on the segment $[AD]$ and $I \ne J$, we always have $\angle JBI > \angle JCI$. c) Let $ABC$ be a triangle with $AB < AC$ and angle bisector $AD$. Choose $M, N$ on segments $CD$ and $BD$, respectively, such that $AD$ is the bisector of angle $\angle MAN$. On the segment $[AD]$ take an arbitrary point $I$ (other than $D$). The lines $BI$, $CI$ intersect $AM$, $AN$ at $B', C'$. Prove that $BB' < CC'$.

In the figure below, circle $O$ has two tangents, $\overline{AC}$ and $\overline{BC}$. $\overline{EF}$ is drawn tangent to circle $O$ such that $E$ is on $\overline{AC}$, $F$ is on $\overline{BC}$, and $\overline{EF} \perp \overline{FC}$. Given that the diameter of circle $O$ has length $10$ and that $CO = 13$, what is the area of triangle $EFC$? [img]https://cdn.artofproblemsolving.com/attachments/b/d/4a1bc818a5e138ae61f1f3d68f6ee5adc1ed6f.png[/img]

It is necessary to construct an angle whose sine is three times greater than its cosine. Describe how this can be done.

The ratio of the sides of a parallelogram is $\lambda>1$. Given $\lambda$, determine the maximum of the acute angle subtended by the diagonals of the parallelogram.

$ABC$ is an isosceles triangle such that $AB = AC$ and $\angle BAC = 96^o$. $D$ is the point for which $\angle ACD = 48^o$, $AD = BC$ and triangle $DAC$ is obtuse-angled. Find $\angle DAC$.

In a matrix $2n \times 2n$, $n \in N$, are $4n^2$ real numbers with a sum equal zero. The absolute value of each of these numbers is not greater than $1$. Prove that the absolute value of a sum of all the numbers from one column or a row doesn't exceed $n$.

In the plane, $2018$ points are given such that all distances between them are different. For each point, mark the closest one of the remaining points. What is the minimal number of marked points?

A fly and an ant are on one corner of a unit cube. They wish to head to the opposite corner of the cube. The fly can fly through the interior of the cube, while the ant has to walk across the faces of the cube. How much shorter is the fly's path if both insects take the shortest path possible?

5. In tetrahedron $ABCD$ following equalities hold: $\angle BAC+\angle BDC=\angle ABD+\angle ACD$ $\angle BAD+\angle BCD=\angle ABC+\angle ADC$ Prove that center of sphere circumscribed about ABCD lies on a line through midpoints of $AB$ and $CD$.

Let $ABC$ be a triangle and $O$ be its circumcenter. Let $D$, $E$ and $F$ be the respective tangent points of the incircle of $\triangle ABC$, and sides $BC$, $CA$ and $AB$. Let $M$ and $N$ be the respective midpoints of sides $AB$ and $AC$. Let $M'$ and $N'$ be the respective reflections of points $M$ and $N$ across lines $DE$ and $DF$. Let lines $CM'$ and $BN'$ intersect lines $DE$ and $DF$ at points $H$ and $J$, respectively. Prove that the points $H$, $J$ and $O$ are collinear. [i]Proposed by Luu Dong, Vietnam[/i]

Let $\alpha,\beta ,\gamma$ be the angles of $\triangle ABC$. a) Show that $cos^2\alpha +cos^2\beta +cos^2 \gamma =1-2cos\alpha cos\beta cos\gamma$ . b) Given that $cos\alpha : cos\beta : cos\gamma = 39 : 33 : 25$, find $sin\alpha : sin\beta : sin\gamma$ .

The perpendicular bisector of $AB$ of an acute $\Delta ABC$ intersects $BC$ and the continuation of $AC$ in points $P$ and $Q$ respectively. $M$ and $N$ are the middle points of side $AB$ and segment $PQ$ respectively. If the lines $AB$ and $CN$ intersect in point $D$, prove that $\Delta ABC$ and $\Delta DCM$ have a common orthocenter.

Frist Campus Center is located $1$ mile north and $1$ mile west of Fine Hall. The area within $5$ miles of Fine Hall that is located north and east of Frist can be expressed in the form $\frac{a}{b} \pi - c$, where $a, b, c$ are positive integers and $a$ and $b$ are relatively prime. Find $a + b + c$.

$\bigtriangleup ABC$ triangle such that $AB = AC, \angle BAC = 20 \textdegree$. $P$ is on $AB$ such that $AP = BC$, find $\frac{1}{2}\angle APC$ in degrees.

In tetrahedron $ ABCD$, $ G_1,G_2$ and $ G_3$ are barycenters of the faces $ ACD,ABD$ and $ BCD$ respectively. (a) Prove that the straight lines $ BG_1,CG_2$ and $ AG_3$ are concurrent. (b) Knowing that $ AG_3\equal{}8,BG_1\equal{}12$ and $ CG_2\equal{}20$ compute the maximum possible value of the volume of $ ABCD$.

Prove that the sum of the distances from any point in space from the vertices of a cube with edge $a$ is not less than $4\sqrt3 a$.

Let ${\cal C}_1$ and ${\cal C}_2$ be concentric circles, with ${\cal C}_2$ in the interior of ${\cal C}_1$. From a point $A$ on ${\cal C}_1$ one draws the tangent $AB$ to ${\cal C}_2$ ($B\in {\cal C}_2$). Let $C$ be the second point of intersection of $AB$ and ${\cal C}_1$, and let $D$ be the midpoint of $AB$. A line passing through $A$ intersects ${\cal C}_2$ at $E$ and $F$ in such a way that the perpendicular bisectors of $DE$ and $CF$ intersect at a point $M$ on $AB$. Find, with proof, the ratio $AM/MC$.

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. [i]Author: Kei Irie, Japan[/i]

Given $n>4$, prove that every cyclic quadrilateral can be dissected into $n$ cyclic quadrilaterals.