In quadrilateral $ABCD$, $m\angle B=m\angle C=120^\circ$, $AB=3$, $BC=4$, and $CD=5$. Find the area of $ABCD$. $\textbf{(A) }15\qquad\textbf{(B) }9\sqrt3\qquad\textbf{(C) }\dfrac{45\sqrt3}4\qquad\textbf{(D) }\dfrac{47\sqrt3}4\qquad\textbf{(E) }15\sqrt3$

Given that points $ D,E$ lie on the sidelines $ AB,BC$ of triangle $ ABC$, respectively, point $ P$ is in interior of triangle $ ABC$ such that $ PE \equal{} PC$ and $ \bigtriangleup DEP\sim \bigtriangleup PCA.$ Prove that $ BP$ is tangent of the circumcircle of triangle $ PAD.$

Let $< \Gamma _j >$ be a sequnce of concentric circles such that the sequence $< R_j >$ , where $R_j$ denotes the radius of $\Gamma_j$, is increasing and $R_j \longrightarrow \infty$ as $j \longrightarrow \infty$. Let $A_1 B_1 C_1$ be a triangle inscribed in $\Gamma _1$. extend the rays $\vec{A_i B_1} , \vec{B_1 C_1 }, \vec{C_1 A_1}$ to meet $\Gamma_2$ in $B_2, C_2$and $A_2$ respectively and form the triangle $A_2 B_2 C_2$. Continue this process. Show that the sequence of triangles $< A_n B_n C_n >$ tends to an equilateral triangle as $n \longrightarrow \infty$

The bisector $AD$ is drawn in the triangle $ABC$. Circle $k$ passes through the vertex $A$ and touches the side $BC$ at point $D$. Prove that the circle circumscribed around $ABC$ touches the circle $k$ at point $A$.

Four congruent semicircular half-disks are arranged inside a circle with radius $4$ so that each semicircle is internally tangent to the circle, and the diameters of the semicircles form a $2\times 2$ square centered at the center of the circle as shown. The radius of each semicircular half-disk is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/f/e/8c0b9fdd69f6b54d39708da94ef2b2d039cb1e.png[/img]

Let $ABC$ be a triangle with $\angle A = 90^{\circ}$. Points $D$ and $E$ lie on sides $AC$ and $AB$, respectively, such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$. Segments $BD$ and $CE$ meet at $I$. Determine whether or not it is possible for segments $AB$, $AC$, $BI$, $ID$, $CI$, $IE$ to all have integer lengths.

Let $ABC$ be an acute triangle such that $AB < AC$. Let $\omega$ be the circumcircle of $ABC$ and assume that the tangent to $\omega$ at $A$ intersects the line $BC$ at $D$. Let $\Omega$ be the circle with center $D$ and radius $AD$. Denote by $E$ the second intersection point of $\omega$ and $\Omega$. Let $M$ be the midpoint of $BC$. If the line $BE$ meets $\Omega$ again at $X$, and the line $CX$ meets $\Omega$ for the second time at $Y$, show that $A, Y$, and $M$ are collinear. [i]Proposed by Nikola Velov, North Macedonia[/i]

The $A$-excircle of a triangle $ABC$ touches the side $BC$ at the point $K$ and the extended side $AB$ at the point $L$. The $B$-excircle touches the lines $BA$ and $BC$ at the points $M$ and $N$, respectively. The lines $KL$ and $MN$ meet at the point $X$. Show that the line $CX$ bisects the angle $ACN$.

In rectangle $ABCD$, $AB=6$ and $AD=8$. Point $M$ is the midpoint of $\overline{AD}$. What is the area of $\triangle AMC$? $\textbf{(A) }12\qquad\textbf{(B) }15\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad \textbf{(E) }24$

The altitudes of an acute triangle $ABC$ with $AB<AC$ intersect at a point $H$, and $O$ is the center of the circumcircle $\Omega$. The segment $OH$ intersects the circumcircle of $BHC$ at a point $X$, different from $O$ and $H$. The circumcircle of $AOX$ intersects the smaller arc $AB$ of $\Omega$ at point $Y$. Prove that the line $XY$ bisects the segment $BC$. [i]Proposed by A. Tereshin[/i]

let $ABCD$ be a cyclic quadrilateral with $E$,an interior point such that $AB=AD=AE=BC$. Let $DE$ meet the circumcircle of $BEC$ again at $F$. Suppose a common tangent to the circumcircle of $BEC$ and $DEC$ touch the circles at $F$ and $G$ respectively. Show that $GE$ is the external angle bisector of angle $BEF$

In quadrilateral $ABCD$ sides $AD$ and $BC$ aren't parallel. Diagonals $AC$ and $BD$ intersect in $E$. $F$ and $G$ are points on sides $AB$ and $DC$ such $\frac{AF}{FB}=\frac{DG}{GC}=\frac{AD}{BC}$ Prove that if $E, F, G$ are collinear then $ABCD$ is cyclic.

Let $M$ be the intersection of diagonals of the convex quadrilateral $ABCD$, where $m(\widehat{AMB})=60^\circ$. Let the points $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of the triangles $ABM$, $BCM$, $CDM$, $DAM$, respectively. What is $Area(ABCD)/Area(O_1O_2O_3O_4)$? $ \textbf{(A)}\ \dfrac 12 \qquad\textbf{(B)}\ \dfrac 32 \qquad\textbf{(C)}\ \dfrac {\sqrt 3}2 \qquad\textbf{(D)}\ \dfrac {1+2\sqrt 3}2 \qquad\textbf{(E)}\ \dfrac {1+\sqrt 3}2 $

Find the smallest positive integer $n$ such that $\underbrace{2^{2^{...^{2}}}}_{n}> 3^{3^{3^3}}$. (The notation $\underbrace{2^{2^{...^{2}}}}_{n}$ is used to denote a power tower with $n$ $2$’s. For example, $\underbrace{2^{2^{...^{2}}}}_{n}$ with $n = 4$ would equal $2^{2^{2^2}}$.)

Let $ABCD$ be a rectangle with $AB \ne BC$ and the center the point $O$. Perpendicular from $O$ on $BD$ intersects lines $AB$ and $BC$ in points $E$ and $F$ respectively. Points $M$ and $N$ are midpoints of segments $[CD]$ and $[AD]$ respectively. Prove that $FM \perp EN$ .

Given is a triangle $\triangle{ABC}$,with $AB<AC<BC$,inscribed in circle $c(O,R)$.Let $D,E,Z$ be the midpoints of $BC,CA,AB$ respectively,and $K$ the foot of the altitude from $A$.At the exterior of $\triangle{ABC}$ and with the sides $AB,AC$ as diameters,we construct the semicircles $c_1,c_2$ respectively.Suppose that $P\equiv DZ\cap c_1 \ , \ S\equiv KZ\cap c_1$ and $R\equiv DE\cap c_2 \ , \ T\equiv KE\cap c_2$.Finally,let $M$ be the intersection of the lines $PS,RT$. [b]i.[/b] Prove that the lines $PR,ST$ intersect at $A$. [b]ii.[/b] Prove that the lines $PR\cap MD$ intersect on $c$. [asy]import graph; size(8cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.8592569519241255, xmax = 12.331775417316715, ymin = -3.1864435704043403, ymax = 6.540061585876658; /* image dimensions */ pen aqaqaq = rgb(0.6274509803921569,0.6274509803921569,0.6274509803921569); pen uququq = rgb(0.25098039215686274,0.25098039215686274,0.25098039215686274); draw((0.6699432366054657,3.2576036755978928)--(0.,0.)--(5.,0.)--cycle, aqaqaq); /* draw figures */ draw((0.6699432366054657,3.2576036755978928)--(0.,0.), uququq); draw((0.,0.)--(5.,0.), uququq); draw((5.,0.)--(0.6699432366054657,3.2576036755978928), uququq); draw(shift((0.33497161830273287,1.6288018377989464))*xscale(1.662889476749906)*yscale(1.662889476749906)*arc((0,0),1,78.3788505217281,258.3788505217281)); draw(shift((2.834971618302733,1.6288018377989464))*xscale(2.7093067970187343)*yscale(2.7093067970187343)*arc((0,0),1,-36.95500560847834,143.0449943915217)); draw((0.6699432366054657,3.2576036755978928)--(0.6699432366054657,0.)); draw((-0.9938564482532047,2.628510486065423)--(2.5,0.)); draw((0.6699432366054657,0.)--(0.,3.2576036755978923)); draw((0.6699432366054657,0.)--(5.,3.257603675597893)); draw((2.5,0.)--(3.3807330143335355,4.282570444700163)); draw((-0.9938564482532047,2.628510486065423)--(2.5,4.8400585427926455)); draw((2.5,4.8400585427926455)--(5.,3.257603675597893)); draw((-0.9938564482532047,2.628510486065423)--(3.3807330143335355,4.282570444700163), linewidth(1.2) + linetype("2 2")); draw((0.,3.2576036755978923)--(5.,3.257603675597893), linewidth(1.2) + linetype("2 2")); draw(circle((2.5,1.18355242571055), 2.766007292905304), linewidth(0.4) + linetype("2 2")); draw((2.5,4.8400585427926455)--(2.5,0.), linewidth(1.2) + linetype("2 2")); /* dots and labels */ dot((0.6699432366054657,3.2576036755978928),linewidth(3.pt) + dotstyle); label("$A$", (0.7472169504504719,2.65), NE * labelscalefactor); dot((0.,0.),linewidth(3.pt) + dotstyle); label("$B$", (-0.2,-0.4), NE * labelscalefactor); dot((5.,0.),linewidth(3.pt) + dotstyle); label("$C$", (5.028818057451246,-0.34281415594345044), NE * labelscalefactor); dot((2.5,0.),linewidth(3.pt) + dotstyle); label("$D$", (2.4275434226319077,-0.32665717063401356), NE * labelscalefactor); dot((2.834971618302733,1.6288018377989464),linewidth(3.pt) + dotstyle); label("$E$", (3.073822835009383,1.5637101105701008), NE * labelscalefactor); dot((0.33497161830273287,1.6288018377989464),linewidth(3.pt) + dotstyle); label("$Z$", (0.003995626216375389,1.402140257475732), NE * labelscalefactor); dot((0.6699432366054657,0.),linewidth(3.pt) + dotstyle); label("$K$", (0.6179610679749769,-0.3105001853245767), NE * labelscalefactor); dot((-0.9938564482532047,2.628510486065423),linewidth(3.pt) + dotstyle); label("$P$", (-1.0785223895158957,2.7916409940873033), NE * labelscalefactor); dot((0.,3.2576036755978923),linewidth(3.pt) + dotstyle); label("$S$", (-0.14141724156855653,3.454077391774215), NE * labelscalefactor); dot((5.,3.257603675597893),linewidth(3.pt) + dotstyle); label("$T$", (5.061132028070119,3.3571354799175936), NE * labelscalefactor); dot((3.3807330143335355,4.282570444700163),linewidth(3.pt) + dotstyle); label("$R$", (3.445433497126431,4.375025554412117), NE * labelscalefactor); dot((2.5,4.8400585427926455),linewidth(3.pt) + dotstyle); label("$M$", (2.5567993051074027,4.940520040242407), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy]

Let $a, b, c$ be sides of a triangle whose perimeter does not exceed $2 \cdot \pi.$, Prove that $\sin a, \sin b, \sin c$ are sides of a triangle.

A line through point $P$ outside of circle $O$ meets the said circle at $B,C$ ($PB < PC$). Let $PO$ meet circle $O$ at $Q,D$ (with $PQ < PD$). Let the line passing $Q$ and perpendicular to $BC$ meet circle $O$ at $A$. If $BD^2 = AD\cdot CP$, prove that $PA$ is a tangent to $O$.

Let $A_0A_1A_2A_3A_4A_5$ be a convex hexagon inscribed in a circle. Define the points $A_0'$, $A_2'$, $A_4'$ on the circle, such that \[ A_0A_0' \parallel A_2A_4, \quad A_2A_2' \parallel A_4A_0, \quad A_4A_4' \parallel A_2A_0 . \] Let the lines $A_0'A_3$ and $A_2A_4$ intersect in $A_3'$, the lines $A_2'A_5$ and $A_0A_4$ intersect in $A_5'$ and the lines $A_4'A_1$ and $A_0A_2$ intersect in $A_1'$. Prove that if the lines $A_0A_3$, $A_1A_4$ and $A_2A_5$ are concurrent then the lines $A_0A_3'$, $A_4A_1'$ and $A_2A_5'$ are also concurrent.

In the usual coordinate system $ xOy$ a line $ d$ intersect the circles $ C_1:$ $ (x\plus{}1)^2\plus{}y^2\equal{}1$ and $ C_2:$ $ (x\minus{}2)^2\plus{}y^2\equal{}4$ in the points $ A,B,C$ and $ D$ (in this order). It is known that $ A\left(\minus{}\frac32,\frac{\sqrt3}2\right)$ and $ \angle{BOC}\equal{}60^{\circ}$. All the $ Oy$ coordinates of these $ 4$ points are positive. Find the slope of $ d$.

[b]p1.[/b] Twelve people, three of whom are in the Mafia and one of whom is a police inspector, randomly sit around a circular table. What is the probability that the inspector ends up sitting next to at least one of the Mafia? [b]p2.[/b] Of the positive integers between $1$ and $1000$, inclusive, how many of them contain neither the digit “$4$” nor the digit “$7$”? [b]p3.[/b] You are really bored one day and decide to invent a variation of chess. In your variation, you create a new piece called the “krook,” which, on any given turn, can move either one square up or down, or one square left or right. If you have a krook at the bottom-left corner of the chessboard, how many different ways can the krook reach the top-right corner of the chessboard in exactly $17$ moves? [b]p4.[/b] Let $p$ be a prime number. What is the smallest positive integer that has exactly $p$ different positive integer divisors? Write your answer as a formula in terms of $p$. [b]p5.[/b] You make the square $\{(x, y)| - 5 \le x \le 5, -5 \le y \le 5\}$ into a dartboard as follows: (i) If a player throws a dart and its distance from the origin is less than one unit, then the player gets $10$ points. (ii) If a player throws a dart and its distance from the origin is between one and three units, inclusive, then the player gets awarded a number of points equal to the number of the quadrant that the dart landed on. (The player receives no points for a dart that lands on the coordinate axes in this case.) (iii) If a player throws a dart and its distance from the origin is greater than three units, then the player gets $0$ points. If a person throws three darts and each hits the board randomly (i.e with uniform distribution), what is the expected value of the score that they will receive? [b]p6.[/b] Teddy works at Please Forget Meat, a contemporary vegetarian pizza chain in the city of Gridtown, as a deliveryman. Please Forget Meat (PFM) has two convenient locations, marked with “$X$” and “$Y$ ” on the street map of Gridtown shown below. Teddy, who is currently at $X$, needs to deliver an eggplant pizza to $\nabla$ en route to $Y$ , where he is urgently needed. There is currently construction taking place at $A$, $B$, and $C$, so those three intersections will be completely impassable. How many ways can Teddy get from $X$ to $Y$ while staying on the roads (Traffic tickets are expensive!), not taking paths that are longer than necessary (Gas is expensive!), and that let him pass through $\nabla$ (Losing a job is expensive!)? [img]https://cdn.artofproblemsolving.com/attachments/e/0/d4952e923dc97596ad354ed770e80f979740bc.png[/img] [b]p7.[/b] $x, y$, and $z$ are positive real numbers that satisfy the following three equations: $$x +\frac{1}{y}= 4 \,\,\,\,\, y +\frac{1}{z}= 1\,\,\,\,\, z +\frac{1}{x}=\frac73.$$ Compute $xyz$. [b]p8.[/b] Alan, Ben, and Catherine will all start working at the Duke University Math Department on January $1$st, $2009$. Alan’s work schedule is on a four-day cycle; he starts by working for three days and then takes one day off. Ben’s work schedule is on a seven-day cycle; he starts by working for five days and then takes two days off. Catherine’s work schedule is on a ten-day cycle; she starts by working for seven days and then takes three days off. On how many days in $2009$ will none of the three be working? [b]p9.[/b] $x$ and $y$ are complex numbers such that $x^3 + y^3 = -16$ and $(x + y)^2 = xy$. What is the value of $|x + y|$? [b]p10.[/b] Call a four-digit number “well-meaning” if (1) its second digit is the mean of its first and its third digits and (2) its third digit is the mean of its second and fourth digits. How many well-meaning four-digit numbers are there? (For a four-digit number, its first digit is its thousands [leftmost] digit and its fourth digit is its units [rightmost] digit. Also, four-digit numbers cannot have “$0$” as their first digit.) [b]p11.[/b] Suppose that $\theta$ is a real number such that $\sum^{\infty}{k=2} \sin \left(2^k\theta \right)$ is well-defined and equal to the real number $a$. Compute: $$\sum^{\infty}{k=0} \left(\cot^3 \left(2^k\theta \right)-\cot \left(2^k\theta \right) \right) \sin^4 \left(2^k\theta \right).$$ Write your answer as a formula in terms of $a$. [b]p12.[/b] You have $13$ loaded coins; the probability that they come up as heads are $\cos\left( \frac{0\pi}{24 }\right)$,$ \cos\left( \frac{1\pi}{24 }\right)$, $\cos\left( \frac{2\pi}{24 }\right)$, $...$, $\cos\left( \frac{11\pi}{24 }\right)$ and $\cos\left( \frac{12\pi}{24 }\right)$, respectively. You throw all $13$ of these coins in the air at once. What is the probability that an even number of them come up as heads? [b]p13.[/b] Three married couples sit down on a long bench together in random order. What is the probability that none of the husbands sit next to their respective wives? [b]p14.[/b] What is the smallest positive integer that has at least $25$ different positive divisors? [b]p15.[/b] Let $A_1$ be any three-element set, $A_2 = \{\emptyset\}$, and $A_3 = \emptyset$. For each $i \in \{1, 2, 3\}$, let: (i) $B_i = \{\emptyset,A_i\}$, (ii) $C_i$ be the set of all subsets of $B_i$, (iii) $D_i = B_i \cup C_i$, and (iv) $k_i$ be the number of different elements in $D_i$. Compute $k_1k_2k_3$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The radii of the circumscribed circle and the inscribed circle of a regular $n$-gon, $n\ge 3$ are denoted by $R_n$ and $r_n$, respectively. Prove that \[\frac{r_n}{R_n}\ge\left(\frac{r_{n+1}}{R_{n+1}}\right)^2.\]

The intersection of a cube and a plane is a pentagon. Prove the length of at least oneside of the pentagon differs from 1 metre by at least 20 centimetres.

Regular hexagon $NOSAME$ with side length $1$ and square $UDON$ are drawn in the plane such that $UDON$ lies outside of $NOSAME$. Compute $[SAND] + [SEND]$, the sum of the areas of quadrilaterals $SAND$ and $SEND$.

Is there a scalene triangle $ABC$ similar to triangle $IHO$, where $I$, $H$, and $O$ are the incenter, orthocenter, and circumcenter, respectively, of triangle $ABC$? [i]Proposed by Li4 and usjl.[/i]