In triangle $ABC$, $X$ and $Y$ are the tangency points of incircle (with center $I$) with sides $AB$ and $AC$ respectively. A tangent line to the circumcircle of triangle $ABC$ (with center $O$) at point $A$, intersects the extension of $BC$ at $D$. If $D,X$ and $Y$ are collinear then prove that $D,I$ and $O$ are also collinear. [i]proposed by Amirhossein Zabeti[/i]

Each of Mahdi and Morteza has drawn an inscribed $93$-gon. Denote the first one by $A_1A_2…A_{93}$ and the second by $B_1B_2…B_{93}$. It is known that $A_iA_{i+1} // B_iB_{i+1}$ for $1 \le i \le 93$ ($A_{93} = A_1, B_{93} = B_1$). Show that $\frac{A_iA_{i+1} }{ B_iB_{i+1}}$ is a constant number independent of $i$. by Morteza Saghafian

In the acute angle triangle $ABC$ the point $O$ is the center of the circumscribed circle and the lines $OA,OB,OC$ intersect sides $BC,CA,AB$ respectively in points $M,N,P$ such that $\angle NMP=90^o$. [b](a)[/b] Find the ratios $\frac{\angle AMN}{\angle NMC}$,$\frac{\angle AMP}{\angle PMB}$. [b](b)[/b] If any of the angles of the triangle $ABC$ is $60^o$, find the two other angles.

Three non-collinear points $A_1, A_2, A_3$ are given in a plane. For $n = 4, 5, 6, \ldots$, $A_n$ be the centroid of the triangle $A_{n-3}A_{n-2}A_{n-1}$. [list] a) Show that there is exactly one point $S$, which lies in the interior of the triangle $A_{n-3}A_{n-2}A_{n-1}$ for all $n\ge 4$. b) Let $T$ be the intersection of the line $A_1A_2$ with $SA_3$. Determine the two ratios, $A_1T : TA_2$ and $TS : SA_3$. [/list]

$ABC$ is an acute angle triangle such that $AB>AC$ and $\hat{BAC}=60^{\circ}$. Let's denote by $O$ the center of the circumscribed circle of the triangle and $H$ the intersection of altitudes of this triangle. Line $OH$ intersects $AB$ in point $P$ and $AC$ in point $Q$. Find the value of the ration $\frac{PO}{HQ}$.

A circle of radius $5$ is inscribed in a rectangle as shown. The ratio of the the length of the rectangle to its width is $2\ :\ 1$. What is the area of the rectangle? [asy] draw((0,0)--(0,10)--(20,10)--(20,0)--cycle); draw(circle((10,5),5)); [/asy] $ \textbf{(A)}\ 50\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 125\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 200 $

[asy] size(100); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((0,1)--(1,0)); draw((0,0)--(.5,sqrt(3)/2)--(1,0)); label("$A$",(0,0),SW); label("$B$",(1,0),SE); label("$C$",(1,1),NE); label("$D$",(0,1),NW); label("$E$",(.5,sqrt(3)/2),E); label("$F$",intersectionpoint((0,0)--(.5,sqrt(3)/2),(0,1)--(1,0)),2W); //Credit to chezbgone2 for the diagram[/asy] Vertex $E$ of equilateral triangle $ABE$ is in the interior of square $ABCD$, and $F$ is the point of intersection of diagonal $BD$ and line segment $AE$. If length $AB$ is $\sqrt{1+\sqrt{3}}$ then the area of $\triangle ABF$ is $\textbf{(A) }1\qquad\textbf{(B) }\frac{\sqrt{2}}{2}\qquad\textbf{(C) }\frac{\sqrt{3}}{2}$ $\qquad\textbf{(D) }4-2\sqrt{3}\qquad \textbf{(E) }\frac{1}{2}+\frac{\sqrt{3}}{4}$

In this figure $\angle RFS = \angle FDR$, $FD = 4$ inches, $DR = 6$ inches, $FR = 5$ inches, $FS = 7\dfrac{1}{2}$ inches. The length of $RS$, in inches, is: [asy] import olympiad; pair F,R,S,D; F=origin; R=5*dir(aCos(9/16)); S=(7.5,0); D=4*dir(aCos(9/16)+aCos(1/8)); label("$F$",F,SW);label("$R$",R,N); label("$S$",S,SE); label("$D$",D,W); label("$7\frac{1}{2}$",(F+S)/2.5,SE); label("$4$",midpoint(F--D),SW); label("$5$",midpoint(F--R),W); label("$6$",midpoint(D--R),N); draw(F--D--R--F--S--R); markscalefactor=0.1; draw(anglemark(S,F,R)); draw(anglemark(F,D,R)); //Credit to throwaway1489 for the diagram[/asy] $\textbf{(A)}\ \text{undetermined} \qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\dfrac{1}{2} \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 6\dfrac{1}{4}$

The width of rectangle $ABCD$ is twice its height, and the height of rectangle $EFCG$ is twice its width. The point $E$ lies on the diagonal $BD$. Which fraction of the area of the big rectangle is that of the small one? [img]https://1.bp.blogspot.com/-aeqefhbBh5E/XzcBjhgg7sI/AAAAAAAAMXM/B0qSgWDBuqc3ysd-mOitP1LarOtBdJJ3gCLcBGAsYHQ/s0/2004%2BMohr%2Bp1.png[/img]

A bag initially contains red marbles and blue marbles only, with more blue than red. Red marbles are added to the bag until only $ 1/3$ of the marbles in the bag are blue. Then yellow marbles are added to the bag until only $ 1/5$ of the marbles in the bag are blue. Finally, the number of blue marbles in the bag is doubled. What fraction of the marbles now in the bag are blue? $ \textbf{(A)}\ \frac {1}{5}\qquad \textbf{(B)}\ \frac {1}{4}\qquad \textbf{(C)}\ \frac {1}{3}\qquad \textbf{(D)}\ \frac {2}{5}\qquad \textbf{(E)}\ \frac {1}{2}$

When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio $2$ to $1$. How many integer values of $k$ are there such that $0<k\leq 2007$ and the area between the parabola $y=k-x^2$ and the $x$-axis is an integer? [asy] import graph; size(300); defaultpen(linewidth(0.8)+fontsize(10)); real k=1.5; real endp=sqrt(k); real f(real x) { return k-x^2; } path parabola=graph(f,-endp,endp)--cycle; filldraw(parabola, lightgray); draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0)); label("Region I", (0,2*k/5)); label("Box II", (51/64*endp,13/16*k)); label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2)); [/asy]

Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $AC=15$. Let $M$ be the midpoint of $BC$ and let $\Gamma$ be the circle passing through $A$ and tangent to line $BC$ at $M$. Let $\Gamma$ intersect lines $AB$ and $AC$ at points $D$ and $E$, respectively, and let $N$ be the midpoint of $DE$. Suppose line $MN$ intersects lines $AB$ and $AC$ at points $P$ and $O$, respectively. If the ratio $MN:NO:OP$ can be written in the form $a:b:c$ with $a,b,c$ positive integers satisfying $\gcd(a,b,c)=1$, find $a+b+c$. [i]James Tao[/i]

The incircle of the triangle $ABC$ touches the side $AC$ at the point $D$. Another circle passes through $D$ and touches the rays $BC$ and $BA$, the latter at the point $A$. Determine the ratio $\frac{AD}{DC}$.

Let $ABCD$ be a parallelogram with $AB=8$, $AD=11$, and $\angle BAD=60^\circ$. Let $X$ be on segment $CD$ with $CX/XD=1/3$ and $Y$ be on segment $AD$ with $AY/YD=1/2$. Let $Z$ be on segment $AB$ such that $AX$, $BY$, and $DZ$ are concurrent. Determine the area of triangle $XYZ$.

Given a triangle $ABC$, let $A',B',C'$ be the perpendicular feet dropped from the centroid $G$ of the triangle $ABC$ onto the sides $BC,CA,AB$ respectively. Reflect $A',B',C'$ through $G$ to $A'',B'',C''$ respectively. Prove that the lines $AA'',BB'',CC''$ are concurrent.

Let $P$ be an interior point of the triangle $ABC$ which is not on the median belonging to $BC$ and satisfying $\angle CAP = \angle BCP. \: BP \cap CA = \{B'\} \: , \: CP \cap AB = \{C'\}$ and $Q$ is the second point of intersection of $AP$ and the circumcircle of $ABC. \: B'Q$ intersects $CC'$ at $R$ and $B'Q$ intersects the line through $P$ parallel to $AC$ at $S.$ Let $T$ be the point of intersection of lines $B'C'$ and $QB$ and $T$ be on the other side of $AB$ with respect to $C.$ Prove that \[\angle BAT = \angle BB'Q \: \Longleftrightarrow \: |SQ|=|RB'| \]

There are $n\ge3$ integers around a circle. We know that for each of these numbers the ratio between the sum of its two neighbors and the number is a positive integer. Prove that the sum of the $n$ ratios is not greater than $3n$.

Two identical jars are filled with alcohol solutions, the ratio of the volume of alcohol to the volume of water being $p : 1$ in one jar and $q : 1$ in the other jar. If the entire contents of the two jars are mixed together, the ratio of the volume of alcohol to the volume of water in the mixture is $\textbf{(A) }\frac{p+q}{2}\qquad\textbf{(B) }\frac{p^2+q^2}{p+q}\qquad\textbf{(C) }\frac{2pq}{p+q}\qquad\textbf{(D) }\frac{2(p^2+pq+q^2)}{3(p+q)}\qquad\textbf{(E) }\frac{p+q+2pq}{p+q+2}$

Triangle $ABC$ is inscribed in circle $\omega$ with $AB = 5$, $BC = 7$, and $AC = 3$. The bisector of angle $A$ meets side $BC$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $DE$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = \frac mn$, where m and n are relatively prime positive integers. Find $m + n$.

What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594? $\textbf{(A)}\ 110 \qquad \textbf{(B)}\ 165 \qquad \textbf{(C)}\ 330 \qquad \textbf{(D)}\ 625 \qquad \textbf{(E)}\ 660$

In $\triangle RED, RD =1, \angle DRE = 75^\circ$ and $\angle RED = 45^\circ$. Let $M$ be the midpoint of segment $\overline{RD}$. Point $C$ lies on side $\overline{ED}$ such that $\overline{RC} \perp \overline{EM}$. Extend segment $\overline{DE}$ through $E$ to point $A$ such that $CA = AR$. Then $AE = \tfrac{a-\sqrt{b}}{c},$ where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer. Find $a+b+c$.

Three points $A,B,C$ are such that $B\in AC$. On one side of $AC$, draw the three semicircles with diameters $AB,BC,CA$. The common interior tangent at $B$ to the first two semicircles meets the third circle $E$. Let $U,V$ be the points of contact of the common exterior tangent to the first two semicircles. Evaluate the ratio $R=\frac{[EUV]}{[EAC]}$ as a function of $r_{1} = \frac{AB}{2}$ and $r_2 = \frac{BC}{2}$, where $[X]$ denotes the area of polygon $X$.

Let $M$, $N$ and $P$ be midpoints of sides $BC, AC$ and $AB$, respectively, and let $O$ be circumcenter of acute-angled triangle $ABC$. Circumcircles of triangles $BOC$ and $MNP$ intersect at two different points $X$ and $Y$ inside of triangle $ABC$. Prove that \[\angle BAX=\angle CAY.\]

Logan is constructing a scaled model of his town. The city's water tower stands $ 40$ meters high, and the top portion is a sphere that holds $ 100,000$ liters of water. Logan's miniature water tower holds $ 0.1$ liters. How tall, in meters, should Logan make his tower? $ \textbf{(A)}\ 0.04\qquad \textbf{(B)}\ \frac{0.4}{\pi}\qquad \textbf{(C)}\ 0.4\qquad \textbf{(D)}\ \frac{4}{\pi}\qquad \textbf{(E)}\ 4$

Let $E$ and $F$ be the intersections of opposite sides of a convex quadrilateral $ABCD$. The two diagonals meet at $P$. Let $O$ be the foot of the perpendicular from $P$ to $EF$. Show that $\angle BOC=\angle AOD$.