Circles $ w_{1}$ and $ w_{2}$ with centres $ O_{1}$ and $ O_{2}$ are externally tangent at point $ D$ and internally tangent to a circle $ w$ at points $ E$ and $ F$ respectively. Line $ t$ is the common tangent of $ w_{1}$ and $ w_{2}$ at $ D$. Let $ AB$ be the diameter of $ w$ perpendicular to $ t$, so that $ A, E, O_{1}$ are on the same side of $ t$. Prove that lines $ AO_{1}$, $ BO_{2}$, $ EF$ and $ t$ are concurrent.

Let $ABCD$ be a square of side length $13$. Let $E$ and $F$ be points on rays $AB$ and $AD$ respectively, so that the area of square $ABCD$ equals the area of triangle $AEF$. If $EF$ intersects $BC$ at $X$ and $BX = 6$, determine $DF$.

The circles $ C_{1}$ and $ C_{2}$ touch externally at $ M$ and the radius of $ C_{2}$ is larger than that of $ C_{1}$. $ A$ is any point on $ C_{2}$ which does not lie on the line joining the centers of the circles. $ B$ and $ C$ are points on $ C_{1}$ such that $ AB$ and $ AC$ are tangent to $ C_{1}$. The lines $ BM$, $ CM$ intersect $ C_{2}$ again at $ E$, $ F$ respectively. $ D$ is the intersection of the tangent at $ A$ and the line $ EF$. Show that the locus of $ D$ as $ A$ varies is a straight line.

Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths $\sqrt{21}$ and $\sqrt{31}$. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. A parallelepiped is a solid with six parallelogram faces such as the one shown below. [asy] unitsize(2cm); pair o = (0, 0), u = (1, 0), v = 0.8*dir(40), w = dir(70); draw(o--u--(u+v)); draw(o--v--(u+v), dotted); draw(shift(w)*(o--u--(u+v)--v--cycle)); draw(o--w); draw(u--(u+w)); draw(v--(v+w), dotted); draw((u+v)--(u+v+w)); [/asy]

Let $ABCD$ be a square, and let $E$ and $F$ be points on sides $\overline{AB}$ and $\overline{CD}$, respectively, such that $AE:EB=AF:FD=2:1$. Let $G$ be the intersection of $\overline{AF}$ and $\overline{DE}$, and let $H$ be the intersection of $\overline{BF}$ and $\overline{CE}$. Find the ratio of the area of quadrilateral $EGFH$ to the area of square $ABCD$. [asy] size(5cm,0); draw((0,0)--(3,0)); draw((3,0)--(3,3)); draw((3,3)--(0,3)); draw((0,3)--(0,0)); draw((0,0)--(2,3)); draw((1,0)--(3,3)); draw((0,3)--(1,0)); draw((2,3)--(3,0)); label("$A$",(0,3),NW); label("$B$",(3,3),NE); label("$C$",(3,0),SE); label("$D$",(0,0),SW); label("$E$",(2,3),N); label("$F$",(1,0),S); label("$G$",(0.66666667,1),E); label("$H$",(2.33333333,2),W); [/asy]

In parallelogram $ABCD,$ let $O$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}.$ Angles $CAB$ and $DBC$ are each twice as large as angle $DBA,$ and angle $ACB$ is $r$ times as large as angle $AOB.$ Find the greatest integer that does not exceed $1000r.$

$A,B,C$ are on circle $\mathcal C$. $I$ is incenter of $ABC$ , $D$ is midpoint of arc $BAC$. $W$ is a circle that is tangent to $AB$ and $AC$ and tangent to $\mathcal C$ at $P$. ($W$ is in $\mathcal C$) Prove that $P$ and $I$ and $D$ are on a line.

Find the ratio of two numbers if the ratio of their arithmetic mean to their geometric mean is $25 : 24$

Let $ k$ be the inscribed circle of non-isosceles triangle $ \triangle ABC$, which center is $ S$. Circle $ k$ touches sides $ BC,CA,AB$ in points $ P,Q,R$ respectively. Line $ QR$ intersects $ BC$ in point $ M$. Let a circle which contains points $ B$ and $ C$ touch $ k$ in point $ N$. Circumscribed circle of $ \triangle MNP$ intersects line $ AP$ in point $ L$, different from $ P$. Prove that points $ S,L$ and $ M$ are collinear.

A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum-shaped solid $F$, in such a way that the ratio between the areas of the painted surfaces of $C$ and $F$ and the ratio between the volumes of $C$ and $F$ are both equal to $k$. Given that $k=m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Given an acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$. Let $K$ be a point inside $ABC$ which is not $O$ nor $H$. Point $L$ and $M$ are located outside the triangle $ABC$ such that $AKCL$ and $AKBM$ are parallelogram. At last, let $BL$ and $CM$ intersects at $N$, and let $J$ be the midpoint of $HK$. Show that $KONJ$ is also a parallelogram. [i]Raja Oktovin, Pekanbaru[/i]

Mr. J left his entire estate to his wife, his daughter, his son, and the cook. His daughter and son got half the estate, sharing in the ratio of $ 4$ to $ 3$. His wife got twice as much as the son. If the cook received a bequest of $ \$500$, then the entire estate was: $ \textbf{(A)}\ \$3500 \qquad\textbf{(B)}\ \$5500 \qquad\textbf{(C)}\ \$6500 \qquad\textbf{(D)}\ \$7000 \qquad\textbf{(E)}\ \$7500$

Square $ABCD$ is inscribed in a circle. Square $EFGH$ has vertices $E$ and $F$ on $\overline{CD}$ and vertices $G$ and $H$ on the circle. The ratio of the area of square $EFGH$ to the area of square $ABCD$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers and $m<n$. Find $10n+m$.

Let $ A,B$ be different points on a parabola. Prove that we can find $ P_1,P_2,\dots,P_{n}$ between $ A,B$ on the parabola such that area of the convex polygon $ AP_1P_2\dots P_nB$ is maximum. In this case prove that the ratio of $ S(AP_1P_2\dots P_nB)$ to the sector between $ A$ and $ B$ doesn't depend on $ A$ and $ B$, and only depends on $ n$.

The length of the altitude of equilateral triangle $ ABC$ is $3$. A circle with radius $2$, which is tangent to $ \left[BC\right]$ at its midpoint, meets other two sides. If the circle meets $ AB$ and $ AC$ at $ D$ and $ E$, at the outer of $\triangle ABC$ , find the ratio $ \frac {Area\, \left(ABC\right)}{Area\, \left(ADE\right)}$. $\textbf{(A)}\ 2\left(5 \plus{} \sqrt {3} \right) \qquad\textbf{(B)}\ 7\sqrt {2} \qquad\textbf{(C)}\ 5\sqrt {3} \\ \qquad\textbf{(D)}\ 2\left(3 \plus{} \sqrt {5} \right) \qquad\textbf{(E)}\ 2\left(\sqrt {3} \plus{} \sqrt {5} \right)$

Let $ \overline{AB}$ be a diameter of a circle and $ C$ be a point on $ \overline{AB}$ with $ 2 \cdot AC \equal{} BC$. Let $ D$ and $ E$ be points on the circle such that $ \overline{DC} \perp \overline{AB}$ and $ \overline{DE}$ is a second diameter. What is the ratio of the area of $ \triangle DCE$ to the area of $ \triangle ABD$? [asy]unitsize(2.5cm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=3; pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0); pair D=dir(aCos(C.x)), E=(-D.x,-D.y); draw(A--B--D--cycle); draw(D--E--C); draw(unitcircle,white); drawline(D,C); dot(O); clip(unitcircle); draw(unitcircle); label("$E$",E,SSE); label("$B$",B,E); label("$A$",A,W); label("$D$",D,NNW); label("$C$",C,SW); draw(rightanglemark(D,C,B,2));[/asy]$ \textbf{(A)} \ \frac {1}{6} \qquad \textbf{(B)} \ \frac {1}{4} \qquad \textbf{(C)}\ \frac {1}{3} \qquad \textbf{(D)}\ \frac {1}{2} \qquad \textbf{(E)}\ \frac {2}{3}$

The diagram below shows twelve $30-60-90$ triangles placed in a circle so that the hypotenuse of each triangle coincides with the longer leg of the next triangle. The fourth and last triangle in this diagram are shaded. The ratio of the perimeters of these two triangles can be written as $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [asy] size(200); defaultpen(linewidth(0.8)); pair point=(-sqrt(3),0); pair past,unit; path line; for(int i=0;i<=12;++i) { past = point; line=past--origin; unit=waypoint(line,1/200); point=extension(past,rotate(90,past)*unit,origin,dir(180-30*i)); if (i == 4) { filldraw(origin--past--point--cycle,gray(0.7)); } else if (i==12) { filldraw(origin--past--point--cycle,gray(0.7)); } else { draw(origin--past--point); } } draw(origin--point); [/asy]

In triangle $\vartriangle ABC, D$ and $E$ are midpoints of the sides $BC$ and $AC$, respectively. Lines $AD$ and $BE$ are drawn intersecting at $P$. It turns out that $\angle CAD = 15^o$ and $\angle APB = 60^o$. What is the value of $AB/BC$ ?

On the base $AB$ of the isosceles triangle $ABC$, lies the point $P$ such that $AP : PB = 1 : 2$. Determine the minimum of $\angle ACP$.

The area of $\triangle EBD$ is one third of the area of $3-4-5$ $ \triangle ABC$. Segment $DE$ is perpendicular to segment $AB$. What is $BD$? [asy] unitsize(10mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,0), B=(5,0), C=(1.8,2.4), D=(5-4sqrt(3)/3,0), E=(5-4sqrt(3)/3,sqrt(3)); pair[] ps={A,B,C,D,E}; draw(A--B--C--cycle); draw(E--D); draw(rightanglemark(E,D,B)); dot(ps); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$D$",D,S); label("$E$",E,NE); label("$3$",midpoint(A--C),NW); label("$4$",midpoint(C--B),NE); label("$5$",midpoint(A--B),SW);[/asy] $ \textbf{(A)}\ \frac{4}{3} \qquad \textbf{(B)}\ \sqrt{5} \qquad \textbf{(C)}\ \frac{9}{4} \qquad \textbf{(D)}\ \frac{4\sqrt{3}}{3} \qquad \textbf{(E)}\ \frac{5}{2} $

A calculator is broken so that the only keys that still work are the $ \sin$, $ \cos$, and $ \tan$ buttons, and their inverses (the $ \arcsin$, $ \arccos$, and $ \arctan$ buttons). The display initially shows $ 0$. Given any positive rational number $ q$, show that pressing some finite sequence of buttons will yield the number $ q$ on the display. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.

Let $ABCDEF$ be a convex hexagon. Segments $AE$ and $BF$ intersect at $X$ and segments $BD$ and $CE$ intersect in $Y.$ It's known that $$ \angle XBC = \angle XDE = \angle YAB = \angle YEF = 80^\circ \text{ and } \angle XCB = \angle XED = \angle YBA = \angle YFE = \angle 70^\circ.$$ Let $P$ and $Q$ be such points on line $XY$ that segments $PX$ and $AF$ intersect, segments $QY$ and $CD$ intersect and $\angle APF = \angle CQD = 30 ^\circ.$ Estimate the sum: \[ \frac{BX}{BF} + \frac{BY}{BD} + \frac{EX}{EA} + \frac{EY}{EC} + \frac{PX}{PY} + \frac{QY}{QX} \] [i]Proposed by Gogi Khimshiashvili, Georgia [/i]

Let $C_{1}$ and $C_{2}$ be concentric circles, with $C_{2}$ in the interior of $C_{1}$. From a point $A$ on $C_{1}$, draw the tangent $AB$ to $C_{2}$ $(B \in C_{2})$. Let $C$ be the second point of intersection of $AB$ and $C_{1}$,and let $D$ be the midpoint of $AB$. A line passing through $A$ intersects $C_{2}$ 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 $AM/MC$. This question is taken from Mathematical Olympiad Challenges , the 9-th exercise in 1.3 Power of a Point.

What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours?

Let $n\geq 4$ be a positive integer.Out of $n$ people,each of two individuals play table tennis game(every game has a winner).Find the minimum value of $n$,such that for any possible outcome of the game,there always exist an ordered four people group $(a_{1},a_{2},a_{3},a_{4})$,such that the person $a_{i}$ wins against $a_{j}$ for any $1\leq i<j\leq 4$