Let $ABC$ be a triangle with $\angle BAC = 60^o$. Let $E$ be the point on the side $BC$ , such that $2 \angle BAE = \angle ACB$ . Let $D$ be the second intersection of $AB$ and the circumcircle of the triangle $AEC$ and $P$ be the second intersection of $CD$ and the circumcircle of the triangle $DBE$. Calculate the angle $\angle BAP$.

In triangle $ABC$, the angle bisectors $AD$, $BE$ and $CF$ are drawn, intersecting at point $I$. The perpendicular bisector of the segment $AD$ intersects lines $BE$ and $CF$ at points $M$ and $N$, respectively. Prove that points $A$, $I$, $M$ and $ N$ lie on the same circle.

A circle of radius $R$ is divided into $8$ equal parts. The points of division are denoted successively by $A, B, C, D, E, F , G$ and $H$. Find the area of the square formed by drawing the chords $AF$ , $BE$, $CH$ and $DG$.

Suppose $ \vartriangle ABC$ is similar to $\vartriangle DEF$, with $ A$, $ B$, and $C$ corresponding to $D, E$, and $F$ respectively. If $\overline{AB} = \overline{EF}$, $\overline{BC} = \overline{FD}$, and $\overline{CA} = \overline{DE} = 2$, determine the area of $ \vartriangle ABC$.

Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Let $t$ be the amount of time, in seconds, before Jenny and Kenny can see each other again. If $t$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?

Prove that for all natural $n\ge 6 000$ any convex $1994$-gon can be cut into $n$ such quadrilaterals thata circle can be circumscribed around each of them

$12.$ In a rectangle $ABCD$ $AB=8$ and $BC=20.$ Let $P$ be a point on $AD$ such that $\angle{BPC}=90^o.$ If $r_1,r_2,r_3.$ are the radii of the incircles of triangles $APB,$ $BPC,$ and $CPD.$ what is the value of $r_1+r_2+r_3 ?$

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

In triangle $ABC$, let $D$ lie on $AB$ such that $AD = AC$ and $\angle ADC = 20^{\circ}$. Let $l$ be a line through $B$ parallel to $CD$. Let $E$ lie on $l$ with $BE = AD$ so that $AE$ intersects segment $BC$ at $F$. If $\angle ABC = 10^{\circ}$, find the degree measure of $\angle FDC$.

Let $P$ be a point on the circumcircle of triangle $ABC$. Let $A_1$ be the reflection of the orthocenter of triangle $PBC$ about the reflection of the perpendicular bisector of $BC$. Points $B_1$ and $C_1$ are defined similarly. Prove that $A_1,B_1,C_1$ are collinear.

Straight lines $OA$ and $OB$ are perpendicular. Find the locus of endpoints $M$ of all broken lines $OM$ of length $\ell$ which intersect each line parallel to $OA$ or $OB$ at not more than one point.

All six sides of a rectangular solid were rectangles. A one-foot cube was cut out of the rectangular solid as shown. The total number of square feet in the surface of the new solid is how many more or less than that of the original solid? [asy] unitsize(20); draw((0,0)--(1,0)--(1,3)--(0,3)--cycle); draw((1,0)--(1+9*sqrt(3)/2,9/2)--(1+9*sqrt(3)/2,15/2)--(1+5*sqrt(3)/2,11/2)--(1+5*sqrt(3)/2,9/2)--(1+2*sqrt(3),4)--(1+2*sqrt(3),5)--(1,3)); draw((0,3)--(2*sqrt(3),5)--(1+2*sqrt(3),5)); draw((1+9*sqrt(3)/2,15/2)--(9*sqrt(3)/2,15/2)--(5*sqrt(3)/2,11/2)--(5*sqrt(3)/2,5)); draw((1+5*sqrt(3)/2,9/2)--(1+2*sqrt(3),9/2)); draw((1+5*sqrt(3)/2,11/2)--(5*sqrt(3)/2,11/2)); label("$1'$",(.5,0),S); label("$3'$",(1,1.5),E); label("$9'$",(1+9*sqrt(3)/4,9/4),S); label("$1'$",(1+9*sqrt(3)/4,17/4),S); label("$1'$",(1+5*sqrt(3)/2,5),E);label("$1'$",(1/2+5*sqrt(3)/2,11/2),S); [/asy] $\text{(A)}\ 2\text{ less} \qquad \text{(B)}\ 1\text{ less} \qquad \text{(C)}\ \text{the same} \qquad \text{(D)}\ 1\text{ more} \qquad \text{(E)}\ 2\text{ more}$

In the figure below, $AB = 15$, $BD = 18$, $AF = 15$, $DF = 12$, $BE = 24$, and $CF = 17$. Find $BG : FG$. [img]https://cdn.artofproblemsolving.com/attachments/9/e/dc171c52961442f9846d2fce858937ff9fb7e8.png[/img]

Circles ${\omega_1}$ , ${\omega_2}$ are externally tangent at point M and tangent internally with circle ${\omega_3}$ at points ${K}$ and $L$ respectively. Let ${A}$ and ${B}$ be the points that their common tangent at point ${M}$ of circles ${\omega_1}$ and ${\omega_2}$ intersect with circle ${\omega_3.}$ Prove that if ${\angle KAB=\angle LAB}$ then the segment ${AB}$ is diameter of circle ${\omega_3.}$ Theoklitos Paragyiou (Cyprus)

Given three congruent rectangles in the space. Their centers coincide, but the planes they lie in are mutually perpendicular. For any two of the three rectangles, the line of intersection of the planes of these two rectangles contains one midparallel of one rectangle and one midparallel of the other rectangle, and these two midparallels have different lengths. Consider the convex polyhedron whose vertices are the vertices of the rectangles. [b]a.)[/b] What is the volume of this polyhedron ? [b]b.)[/b] Can this polyhedron turn out to be a regular polyhedron ? If yes, what is the condition for this polyhedron to be regular ?

Let $ ABC$ be a triangle and $ A'$ , $ B'$ and $ C'$ lie on $ BC$ , $ CA$ and $ AB$ respectively such that the incenter of $ A'B'C'$ and $ ABC$ are coincide and the inradius of $ A'B'C'$ is half of inradius of $ ABC$ . Prove that $ ABC$ is equilateral .

Given $ABC$ triangle with incircle $L_1$ and circumcircle $L_2$. If points $X, Y, Z$ lie on $L_2$, such that $XY, XZ$ are tangent to $L_1$, then prove that $YZ$ is also tangent to $L_1$.

In an acute-angled triangle $ABC$ the interior bisector of angle $A$ meets $BC$ at $L$ and meets the circumcircle of $ABC$ again at $N$. From $L$ perpendiculars are drawn to $AB$ and $AC$, with feet $K$ and $M$ respectively. Prove that the quadrilateral $AKNM$ and the triangle $ABC$ have equal areas.[i](IMO Problem 2)[/i] [i]Proposed by Soviet Union.[/i]

Call the ordered pair of distinct circles $(\omega, \gamma)$ scribable if there exists a triangle with circumcircle $\omega$ and incircle $\gamma$. Prove that among $n$ distinct circles there are at most $(n/2)^2$ scribable pairs. [i]Proposed by Daniel Liu

In the Quadrilateral $ABCD$ we have $ \measuredangle B=\measuredangle D = 60^\circ $.$M$ is midpoint of side $AD$.The line through $M$ parallel to $CD$ meets $BC$ at $P$.Point $X$ lying on $CD$ such that $BX=MX$.Prove that $AB=BP$ if and only if $\measuredangle MXB=60^\circ$. Author: Davoud Vakili, Iran

Given a square $ABCD$ of side $a$, we consider the circle $\omega$, tangent to side $BC$ and to the two semicircles of diameters $AB$ and $CD$. Calculate the radius of circle $\omega$,

You have a paper pentagon, $ABCDE$, such that $AB = BC = 3$ cm, $CD = DE= 5$ cm, $EA = 4$ cm, $\angle ABC = 100^o$ ,$ \angle CDE = 80^o$. You have to divide the pentagon into four triangles, by three straight cuts, so that with the four triangles assemble a rectangle, without gaps or overlays. (The triangles can be rotated and / or turned around.)

In a non isoceles triangle $ABC$, let the perpendicular bisector of $[BC]$ intersect $(ABC)$ at $M$ and $N$ respectively. Let the midpoints of $[AM]$ and $[AN]$ be $K$ and $L$ respectively. Let $(ABK)$ and $(ABL)$ intersect $AC$ again at $D$ and $E$ respectively, let $(ACK)$ and $(ACL)$ intersect $AB$ again at $F$ and $G$ respectively. Prove that the lines $DF$, $EG$ and $MN$ are concurrent.

On the circumcircle of triangle $ABC$, point $P$ is chosen, such that the perpendicular drawn from point $P$ to line $AC$ intersects the circle again at a point $Q$, the perpendicular drawn from point $Q$ to line $AB$ intersects the circle again at a point $R$ and the perpendicular drawn from point $R$ to line $BC$ intersects the circle again at the initial point $P$. Let $O$ be the centre of this circle. Prove that $\angle POC = 90^o$.

In triangle $ABC$, $AD$ and $AH$ are the angle bisector and the altitude of vertex $A$, respectively. The perpendicular bisector of $AD$, intersects the semicircles with diameters $AB$ and $AC$ which are drawn outside triangle $ABC$ in $X$ and $Y$, respectively. Prove that the quadrilateral $XYDH$ is concyclic. [i]Proposed by Mahan Malihi[/i]