There are a number of arcs on the edge of a circular disk. Each pair of arcs has the least one point in common. Show that on the circle you can choose two diametrical opposites points such that each arc contains at least one of these two points.

Andy's lawn has twice as much area as Beth's lawn and three times as much area as Carlos' lawn. Carlos' lawn mower cuts half as fast as Beth's mower and one third as fast as Andy's mower. If they all start to mow their lawns at the same time, who will finish first? $ \textbf{(A)}\ \text{Andy} \qquad \textbf{(B)}\ \text{Beth} \qquad \textbf{(C)}\ \text{Carlos} \qquad \textbf{(D)}\ \text{Andy and Carlos tie for first.}$ $\textbf{(E)}\ \text{All three tie.}$

Let $A_0B_0C_0$ be a triangle. For a positive integer $n \geq 1$, we define $A_n$ on the segment $B_{n-1}C_{n-1}$ such that $B_{n-1}A_n:C_{n-1}A_n=2:1$ and $B_n, C_n$ are defined cyclically in a similar manner. Show that there exists an unique point $P$ that lies in the interior of all triangles $A_nB_nC_n$.

There are two equal circles of radius $1$ placed inside the triangle $ABC$ with side $BC = 6$. The circles are tangent to each other, one is inscribed in angle $B$, the other one is inscribed in angle $C$. (a) Prove that the centroid $M$ of the triangle $ABC$ does not lie inside any of the given circles. (b) Prove that if $M$ lies on one of the circles, then the triangle $ABC$ is isosceles.

In triangle $ABC$, $\angle{BAC}=60^\circ$ and the incircle of $ABC$ touches $AB$ and $AC$ at $P$ and $Q$, respectively. Lines $PC$ and $QB$ intersect at $G$. Let $R$ be the circumradius of $BGC$. Find the minimum value of $R/BC$. [i]Author: Alex Song[/i]

In a square $ABCD$ with side $k$, let $P$ and $Q$ in $BC$ and $DC$ respectively, where $PC = 3PB$ and $QD = 2QC$. Let $M$ be the point of intersection of the lines $AQ$ and $PD$, determine the area of $QMD$ in function of $k$

In an acute triangle $ABC$ the angle $\angle C$ is greater than $\angle A$. Let $E$ be such that $AE$ is a diameter of the circumscribed circle $\Gamma$ of \vartriangle ABC. Let $K$ be the intersection of $AC$ and the tangent line at $B$ to $\Gamma$. Let $L$ be the orthogonal projection of $K$ on $AE$ and let $D$ be the intersection of $KL$ and $AB$. Prove that $CE$ is the bisector of $\angle BCD$.

The tangent lines from the point $A$ to the circle $C$ touches the circle at $M$ and $N$. Let $P$ a point on $[AN]$. Let $MP$ meet $C$ at $Q$. Let $MN$ meet the line through $P$ and parallel to $MA$ at $R$. If $|MA|=2$, $|MN|=\sqrt 3$, and $QR \parallel AN$, what is $|PN|$? $ \textbf{(A)}\ \dfrac 32 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \dfrac {\sqrt 3} 2 \qquad\textbf{(D)}\ \sqrt 2 \qquad\textbf{(E)}\ \sqrt 3 $

Prove that if $a,b,c,d$ are lengths of the successive sides of a quadrangle (not necessarily convex) with the area equal to $S$, then the following inequality holds \[S \leq \frac{1}{2}(ac+bd).\] For which quadrangles does the inequality become equality?

The vertex $E$ of a square $EFGH$ is at the center of square $ABCD$. The length of a side of $ABCD$ is $1$ and the length of a side of $EFGH$ is $2$. Side $EF$ intersects $CD$ at $I$ and $EH$ intersects $AD$ at $J$. If angle $EID=60^\circ$, the area of quadrilateral $EIDJ$ is $\textbf{(A) }\dfrac14\qquad\textbf{(B) }\dfrac{\sqrt3}6\qquad\textbf{(C) }\dfrac13\qquad\textbf{(D) }\dfrac{\sqrt2}4\qquad\textbf{(E) }\dfrac{\sqrt3}2$

There are one-way flights between $100$ cities of a country. It is possible to fly starting from the capital city and visiting all other $99$ cities and returning again to the capital city. Let $ N$ be the smallest number of flights inorder to form such a flight combination. Among all flight combinations (satisfying previous condtions), $ N$ can be at most ? $\textbf{(A)}\ 1850 \qquad\textbf{(B)}\ 2100 \qquad\textbf{(C)}\ 2550 \qquad\textbf{(D)}\ 3060 \qquad\textbf{(E)}\ \text{None}$

A big circle is inscribed in a rhombus, each of two smaller circles touches two sides of the rhombus and the big circle as shown in the figure on the right. Prove that the four dashed lines spanning the points where the circles touch the rhombus as shown in the figure make up a square.

The convex polygon $A_{1}A_{2}\ldots A_{2006}$ has opposite sides parallel $(A_{1}A_{2}||A_{1004}A_{1005}, \ldots)$. Prove that the diagonals $A_{1}A_{1004}, A_{2}A_{1005}, \ldots A_{1003}A_{2006}$ are concurrent if and only if opposite sides are equal.

Let be a convex quadrilateral $ ABCD. $ On the semi-straight line extension of $ AB $ in the direction of $ B, $ put $ A_1 $ such that $ AB=BA_1. $ Similarly, define $ B_1,C_1,D_1, $ for the other three sides. [b]a)[/b] If $ E,E_1,F,F_1 $ are the midpoints of $ BC,A_1B_1,AD $ respectively, $ C_1,D_1, $ show that $ EE_1=FF_1. $ [b]b)[/b] Delete everything, but $ A_1,B_1,C_1,D_1. $ Now, find a way to construct the initial quadrilateral. [i]Vasile Pop[/i]

In the triangle $ABC$ the median $AM$ is drawn. Is it possible that the radius of the circle inscribed in $\vartriangle ABM$ could be twice as large as the radius of the circle inscribed in $\vartriangle ACM$ ? ( D . Fomin , Leningrad)

On a $55\times 55$ square grid, $500$ unit squares were cut out as well as $400$ L-shaped pieces consisting of 3 unit squares (each piece can be oriented in any way) [refer to the figure]. Prove that at least two of the cut out pieces bordered each other before they were cut out. [asy]size(2.013cm); draw ((0,0)--(0,1)); draw ((0,0)--(1,0)); draw ((0,1)--(.5,1)); draw ((.5,1)--(.5,0)); draw ((0,.5)--(1,.5)); draw ((1,.5)--(1,0)); draw ((1,.5)--(1,0)); [/asy]

Let be given a tetrahedron whose circumcenter is $O$. Draw diameters $AA_{1},BB_{1},CC_{1},DD_{1}$ of the circumsphere of $ABCD$. Let $A_{0},B_{0},C_{0},D_{0}$ be the centroids of triangle $BCD,CDA,DAB,ABC$. Prove that $A_{0}A_{1},B_{0}B_{1},C_{0}C_{1},D_{0}D_{1}$ are concurrent at a point, say, $F$. Prove that the line through $F$ and a midpoint of a side of $ABCD$ is perpendicular to the opposite side.

A square piece of paper has sides of length $ 100$. From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at distance $ \sqrt {17}$ from the corner, and they meet on the diagonal at an angle of $ 60^\circ$ (see the figure below). The paper is then folded up along the lines joining the vertices of adjacent cuts. When the two edges of a cut meet, they are taped together. The result is a paper tray whose sides are not at right angles to the base. The height of the tray, that is, the perpendicular distance between the plane of the base and the plane formed by the upper edges, can be written in the form $ \sqrt [n]{m}$, where $ m$ and $ n$ are positive integers, $ m < 1000$, and $ m$ is not divisible by the $ n$th power of any prime. Find $ m \plus{} n$. [asy]import math; unitsize(5mm); defaultpen(fontsize(9pt)+Helvetica()+linewidth(0.7)); pair O=(0,0); pair A=(0,sqrt(17)); pair B=(sqrt(17),0); pair C=shift(sqrt(17),0)*(sqrt(34)*dir(75)); pair D=(xpart(C),8); pair E=(8,ypart(C)); draw(O--(0,8)); draw(O--(8,0)); draw(O--C); draw(A--C--B); draw(D--C--E); label("$\sqrt{17}$",(0,2),W); label("$\sqrt{17}$",(2,0),S); label("cut",midpoint(A--C),NNW); label("cut",midpoint(B--C),ESE); label("fold",midpoint(C--D),W); label("fold",midpoint(C--E),S); label("$30^\circ$",shift(-0.6,-0.6)*C,WSW); label("$30^\circ$",shift(-1.2,-1.2)*C,SSE);[/asy]

In triangle $ABC$, $\angle ABC$ is obtuse. Point $D$ lies on side $AC$ such that $\angle ABD$ is right, and point $E$ lies on side $AC$ between $A$ and $D$ such that $BD$ bisects $\angle EBC$. Find $CE$ given that $AC=35$, $BC=7$, and $BE=5$.

Three disks of diameter $d$ are touching a sphere in their centers. Besides, every disk touches the other two disks. How to choose the radius $R$ of the sphere in order that axis of the whole figure has an angle of $60^\circ$ with the line connecting the center of the sphere with the point of the disks which is at the largest distance from the axis ? (The axis of the figure is the line having the property that rotation of the figure of $120^\circ$ around that line brings the figure in the initial position. Disks are all on one side of the plane, passing through the center of the sphere and orthogonal to the axis).

A regular heptagon $A_1A_2... A_7$ is inscribed in circle $C$. Point $P$ is taken on the shorter arc $A_7A_1$. Prove that $PA_1+PA_3+PA_5+PA_7 = PA_2+PA_4+PA_6$.

The triangle $ABC$ has $CA = CB$. $P$ is a point on the circumcircle between $A$ and $B$ (and on the opposite side of the line $AB$ to $C$). $D$ is the foot of the perpendicular from $C$ to $PB$. Show that $PA + PB = 2 \cdot PD$.

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

On a line a set of segments is given of total length less than $n$. Prove that every set of $n$ points of the line can be translated in some direction along the line for a distance smaller than $\frac{n}{2}$ so that none of the points remain on the segments.