Let the incircle of triangle $ ABC$ touch sides $ BC,\, CA$ and $ AB$ at $ D,\, E$ and $ F,$ respectively. Let $ \omega,\,\omega_{1},\,\omega_{2}$ and $ \omega_{3}$ denote the circumcircles of triangle $ ABC,\, AEF,\, BDF$ and $ CDE$ respectively. Let $ \omega$ and $ \omega_{1}$ intersect at $ A$ and $ P,\,\omega$ and $ \omega_{2}$ intersect at $ B$ and $ Q,\,\omega$ and $ \omega_{3}$ intersect at $ C$ and $ R.$ $ a.$ Prove that $ \omega_{1},\,\omega_{2}$ and $ \omega_{3}$ intersect in a common point. $ b.$ Show that $ PD,\, QE$ and $ RF$ are concurrent.

It is given a right-angled triangle $ABC$ and its circumcircle $k$. (a) prove that the radii of the circle $k_1$ tangent to the cathets of the triangle and to the circle $k$ is equal to the diameter of the incircle of the triangle ABC. (b) on the circle $k$ there may be found a point $M$ for which the sum $MA+MB+MC$ is as large as possible.

Construct a triangle knowing an angle, the ratio of the sides that form it and the radius of the inscribed circle.

Given two points $P,\ Q$ on the parabola $C: y=x^2-x-2$ in the $xy$ plane. Note that the $x$ coodinate of $P$ is less than that of $Q$. (a) If the origin $O$ is the midpoint of the lines　egment $PQ$, then find the equation of the line $PQ$. (b) If the origin $O$ divides internally the line segment $PQ$ by 2:1, then find the equation of $PQ$. (c) If the origin $O$ divides internally the line segment $PQ$ by 2:1, find the area of the figure bounded by the parabola $C$ and the line $PQ$.

In $\triangle ABC,$ a point $E$ is on $\overline{AB}$ with $AE=1$ and $EB=2.$ Point $D$ is on $\overline{AC}$ so that $\overline{DE} \parallel \overline{BC}$ and point $F$ is on $\overline{BC}$ so that $\overline{EF} \parallel \overline{AC}.$ What is the ratio of the area of $CDEF$ to the area of $\triangle ABC?$ [asy] size(7cm); pair A,B,C,DD,EE,FF; A = (0,0); B = (3,0); C = (0.5,2.5); EE = (1,0); DD = intersectionpoint(A--C,EE--EE+(C-B)); FF = intersectionpoint(B--C,EE--EE+(C-A)); draw(A--B--C--A--DD--EE--FF,black+1bp); label("$A$",A,S); label("$B$",B,S); label("$C$",C,N); label("$D$",DD,W); label("$E$",EE,S); label("$F$",FF,NE); label("$1$",(A+EE)/2,S); label("$2$",(EE+B)/2,S); [/asy] $\textbf{(A) } \frac{4}{9} \qquad \textbf{(B) } \frac{1}{2} \qquad \textbf{(C) } \frac{5}{9} \qquad \textbf{(D) } \frac{3}{5} \qquad \textbf{(E) } \frac{2}{3}$

[b]p1.[/b] Trung has $2$ bells. One bell rings $6$ times per hour and the other bell rings $10$ times per hour. At the start of the hour both bells ring. After how much time will the bells ring again at the same time? Express your answer in hours. [b]p2.[/b] In a soccer tournament there are $n$ teams participating. Each team plays every other team once. The matches can end in a win for one team or in a draw. If the match ends with a win, the winner gets $3$ points and the loser gets $0$. If the match ends in a draw, each team gets $1$ point. At the end of the tournament the total number of points of all the teams is $21$. Let $p$ be the number of points of the team in the first place. Find $n + p$. [b]p3.[/b] What is the largest $3$ digit number $\overline{abc}$ such that $b \cdot \overline{ac} = c \cdot \overline{ab} + 50$? [b]p4.[/b] Let s(n) be the number of quadruplets $(x, y, z, t)$ of positive integers with the property that $n = x + y + z + t$. Find the smallest $n$ such that $s(n) > 2014$. [b]p5.[/b] Consider a decomposition of a $10 \times 10$ chessboard into p disjoint rectangles such that each rectangle contains an integral number of squares and each rectangle contains an equal number of white squares as black squares. Furthermore, each rectangle has different number of squares inside. What is the maximum of $p$? [b]p6.[/b] If two points are selected at random from a straight line segment of length $\pi$, what is the probability that the distance between them is at least $\pi- 1$? [b]p7.[/b] Find the length $n$ of the longest possible geometric progression $a_1, a_2,..,, a_n$ such that the $a_i$ are distinct positive integers between $100$ and $2014$ inclusive. [b]p8.[/b] Feng is standing in front of a $100$ story building with two identical crystal balls. A crystal ball will break if dropped from a certain floor $m$ of the building or higher, but it will not break if it is dropped from a floor lower than $m$. What is the minimum number of times Feng needs to drop a ball in order to guarantee he determined $m$ by the time all the crystal balls break? [b]p9.[/b] Let $A$ and $B$ be disjoint subsets of $\{1, 2,..., 10\}$ such that the product of the elements of $A$ is equal to the sum of the elements in $B$. Find how many such $A$ and $B$ exist. [b]p10.[/b] During the semester, the students in a math class are divided into groups of four such that every two groups have exactly $2$ students in common and no two students are in all the groups together. Find the maximum number of such groups. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCDEF$ be a cyclic hexagon such that \[AB=BC,\quad CD=DE,\quad EF=FA.\] Show that \[[ACE]\le[BDF]\] and determine when the equality holds. ($[XYZ]$ denotes the area of the triangle $XYZ.$)

There are five research labs on Mars. Is it always possible to divide Mars to five connected congruent regions such that each region contains exactly on research lab. [img]http://i37.tinypic.com/f2iq8g.png[/img]

The projections of the body on two planes are circles. Prove that they have the same radius.

Points $P_1, P_2, P_3,$ and $P_4$ are $(0,0), (10, 20), (5, 15),$ and $(12, -6)$, respectively. For what point $P \in \mathbb{R}^2$ is the sum of the distances from $P$ to the other $4$ points minimal?

Let $ a,b,c\in\mathbb{C}^* $ pairwise distinct, having the same absolute value, and satisfying: $$ a^2+b^2+c^2-ab-bc-ca=0. $$ Prove that $ a,b,c $ represents the affixes of the vertices of a right or equilateral triangle.

Prove that one can construct two triangles from six edges of an arbitrary tetrahedron. (VV Proizvolov)

Let $ABC$ be an acute triangle with midpoint $M$ of $AB$. The point $D$ lies on the segment $MB$ and $I_1, I_2$ denote the incenters of $\triangle ADC$ and $\triangle BDC$. Given that $\angle I_1MI_2=90^{\circ}$, show that $CA=CB$.

A circle centered at $ O$ has radius $ 1$ and contains the point $ A$. Segment $ AB$ is tangent to the circle at $ A$ and $ \angle{AOB} \equal{} \theta$. If point $ C$ lies on $ \overline{OA}$ and $ \overline{BC}$ bisects $ \angle{ABO}$, then $ OC \equal{}$ [asy]import olympiad; unitsize(2cm); defaultpen(fontsize(8pt)+linewidth(.8pt)); labelmargin=0.2; dotfactor=3; pair O=(0,0); pair A=(1,0); pair B=(1,1.5); pair D=bisectorpoint(A,B,O); pair C=extension(B,D,O,A); draw(Circle(O,1)); draw(O--A--B--cycle); draw(B--C); label("$O$",O,SW); dot(O); label("$\theta$",(0.1,0.05),ENE); dot(C); label("$C$",C,S); dot(A); label("$A$",A,E); dot(B); label("$B$",B,E);[/asy] $ \textbf{(A)}\ \sec^2\theta \minus{} \tan\theta \qquad \textbf{(B)}\ \frac {1}{2} \qquad \textbf{(C)}\ \frac {\cos^2\theta}{1 \plus{} \sin\theta} \qquad \textbf{(D)}\ \frac {1}{1 \plus{} \sin\theta} \qquad \textbf{(E)}\ \frac {\sin\theta}{\cos^2\theta}$

Let $M,N,P$ be the midpoints of the sides $BC, CA, AB$ of a triangle $ABC$. The lines $AM, BN, CP$ intersect the circumcircle of $ABC$ at points $A',B', C'$, respectively. Show that if $A'B'C'$ is an equilateral triangle, then so is $ABC.$

Rectangle $p*q,$ where $p,q$ are relatively coprime positive integers with $p <q$ is divided into squares $1*1$.Diagonal which goes from lowest left vertice to highest right cuts triangles from some squares.Find sum of perimeters of all such triangles.

In triangle $ABC$ with incenter $I$ and circumcenter $O$, let $A',B',C'$ be the points of tangency of its circumcircle with its $A,B,C$-mixtilinear circles, respectively. Let $\omega_A$ be the circle through $A'$ that is tangent to $AI$ at $I$, and define $\omega_B, \omega_C$ similarly. Prove that $\omega_A,\omega_B,\omega_C$ have a common point $X$ other than $I$, and that $\angle AXO = \angle OXA'$. [i]Proposed by Sammy Luo[/i]

In triangle $ABC$, $AD$ and $AE$ trisect $\angle BAC$. The lengths of $BD, DE $ and $EC$ are $1, 3 $ and $5$ respectively. Find the length of $AC$.

Let $ABC$ be an acute-angled triangle in which no two sides have the same length. The reflections of the centroid $G$ and the circumcentre $O$ of $ABC$ in its sides $BC,CA,AB$ are denoted by $G_1,G_2,G_3$ and $O_1,O_2,O_3$, respectively. Show that the circumcircles of triangles $G_1G_2C$, $G_1G_3B$, $G_2G_3A$, $O_1O_2C$, $O_1O_3B$, $O_2O_3A$ and $ABC$ have a common point. [i]The centroid of a triangle is the intersection point of the three medians. A median is a line connecting a vertex of the triangle to the midpoint of the opposite side.[/i]

The square $DEAF$ is constructed inside the $30^o-60^o-90^o$ triangle $ABC$, with the hypotenuse $BC = 4$, $D$ on side $BC$, E on side $AC$, and F on side $AB$. What is the side length of the square?

A square with side $1$ is intersected by two parallel lines as shown in the figure. Find the sum of the perimeters of the shaded triangles if the distance between the lines is also $1$. [img]https://cdn.artofproblemsolving.com/attachments/9/e/4e70610b80871325a72e923a0909eff06aebfa.png[/img]

Given is a acute angled triangle $ABC$. The lengths of the altitudes from $A, B$ and $C$ are successively $h_A, h_B$ and $h_C$. Inside the triangle is a point $P$. The distance from $P$ to $BC$ is $1/3 h_A$ and the distance from $P$ to $AC$ is $1/4 h_B$. Express the distance from $P$ to $AB$ in terms of $h_C$.

Given $ \triangle{ABC}$, where $ A$ is at $ (0,0)$, $ B$ is at $ (20,0)$, and $ C$ is on the positive $ y$-axis. Cone $ M$ is formed when $ \triangle{ABC}$ is rotated about the $ x$-axis, and cone $ N$ is formed when $ \triangle{ABC}$ is rotated about the $ y$-axis. If the volume of cone $ M$ minus the volume of cone $ N$ is $ 140\pi$, find the length of $ \overline{BC}$.

Let $ABC$ be an isosceles triangle, $AB = AC, \angle A = 20^{\circ}$. Let $D$ be a point on $AB$, and $E$ a point on $AC$ such that $\angle ACD = 20^{\circ}$ and $\angle ABE = 30^{\circ}$. What is the measure of the angle $\angle CDE$?

Fix a triangle $ABC$. The variable point $M$ in its interior is such that $\angle MAC = \angle MBC$ and $N$ is the reflection of $M$ with respect to the midpoint of $AB$. Prove that $|AM| \cdot |BM| + |CM| \cdot |CN|$ is independent of the choice of $M$.