Through the circumcenter $O$ of an arbitrary acute-angled triangle, chords $A_1A_2,B_1B_2, C_1C_2$ are drawn parallel to the sides $BC,CA,AB$ of the triangle respectively. If $R$ is the radius of the circumcircle, prove that \[A_1O \cdot OA_2 + B_1O \cdot OB_2 + C_1O \cdot OC_2 = R^2.\]

Find the kind of a triangle if $$\frac{a\cos\alpha+b\cos\beta+c\cos\gamma}{a\sin\alpha+b\sin\beta+c\sin\gamma}=\frac{2p}{9R}.$$ ($\alpha,\beta,\gamma$ are the measures of the angles, $a,b,c$ are the respective lengths of the sides, $p$ the semiperimeter, $R$ is the circumradius) [i]K. Petrov[/i]

Let $ ABCD$ be an isosceles trapezium. Determine the geometric location of all points $ P$ such that \[ |PA| \cdot |PC| \equal{} |PB| \cdot |PD|.\]

Let $\omega$ be the incircle of $\vartriangle ABC$, $\omega$ is tangent to sides $BC$ and $AC$ at $D$ and $E$ respectively. The line perpendicular to $BC$ at $D$ intersects $\omega$ again at $P$. Lines $AP$ and $BC$ intersect at $M$. Let $N$ be a point on segment $AC$ so that $AE = CN$. Line $BN$ intersects $\omega$ at $Q$ (closer to $B$) and intersect $AM$ at $R$. Show that the area of $\vartriangle ABR$ is equal to the area of $PQMN$.

Let $C$ and $D$ be two points on the semicricle with diameter $AB$ such that $B$ and $C$ are on distinct sides of the line $AD$. Denote by $M$, $N$ and $P$ the midpoints of $AC$, $BD$ and $CD$ respectively. Let $O_A$ and $O_B$ the circumcentres of the triangles $ACP$ and $BDP$. Show that the lines $O_AO_B$ and $MN$ are parallel.

Let $ABC$ be an acute, non isosceles with $I$ is its incenter. Denote $D, E$ as tangent points of $(I)$ on $AB,AC$, respectively. The median segments respect to vertex $A$ of triangles $ABE$ and $ACD$ meet$ (I)$ at$ P,Q,$ respectively. Take points $M, N$ on the line $DE$ such that $AM \parallel BE$ and $AN \parallel C D$ respectively. a) Prove that $A$ lies on the radical axis of $(MIP)$ and $(NIQ)$. b) Suppose that the orthocenter $H$ of triangle $ABC$ lies on $(I)$. Prove that there exists a line which is tangent to three circles of center $A, B, C$ and all pass through $H$.

There are $n$ fleas on an infinite sheet of triangulated paper. Initially the fleas are in different small triangles, all of which are inside some equilateral triangle consisting of $n^2$ small triangles. Once a second each flea jumps from its original triangle to one of the three small triangles having a common vertex but no common side with it. For which natural numbers $n$ does there exist an initial configuration such that after a finite number of jumps all the $n$ fleas can meet in a single small triangle?

$ABC$ is a triangle, $G$ its centroid and $A',B',C'$ the midpoints of its sides $BC,CA,AB$, respectively. Prove that if the quadrilateral $AC'GB'$ is cyclic then $AB \cdot CC' = AC \cdot BB'$:

Let $O$ be the circumcentre of the acute triangle $ABC$ and let lines $AO$ and $BC$ intersect at point $K$. On sides $AB$ and $AC$, points $L$ and $M$ are chosen such that $|KL|= |KB|$ and $|KM| = |KC|$. Prove that segments $LM$ and $BC$ are parallel.

[u]Round 1 [/u] [b]p1.[/b] Pirate Jim had $8$ boxes with gun powder weighing $1, 2, 3, 4, 5, 6, 7$, and $8$ pounds (the weight is printed on top of every box). Pirate Bob hid a $1$-pound gold bar in one of these boxes. Pirate Jim has a balance scale that he can use, but he cannot open any of the boxes. Help him find the box with the gold bar using two weighings on the balance scale. [b]p2.[/b] James Bond will spend one day at Dr. Evil's mansion to try to determine the answers to two questions: a) Is Dr. Evil at home? b) Does Dr. Evil have an army of ninjas? The parlor in Dr. Evil's mansion has three windows. At noon, Mr. Bond will sneak into the parlor and use open or closed windows to signal his answers. When he enters the parlor, some windows may already be opened, and Mr. Bond will only have time to open or close one window (or leave them all as they are). Help Mr. Bond and Moneypenny design a code that will tell Moneypenny the answers to both questions when she drives by later that night and looks at the windows. Note that Moneypenny will not have any way to know which window Mr. Bond opened or closed. [b]p3.[/b] Suppose that you have a triangle in which all three side lengths and all three heights are integers. Prove that if these six lengths are all different, there cannot be four prime numbers among them. p4. Fred and George have designed the Amazing Maze, a $5\times 5$ grid of rooms, with Adorable Doors in each wall between rooms. If you pass through a door in one direction, you gain a gold coin. If you pass through the same door in the opposite direction, you lose a gold coin. The brothers designed the maze so that if you ever come back to the room in which you started, you will find that your money has not changed. Ron entered the northwest corner of the maze with no money. After walking through the maze for a while, he had $8$ shiny gold coins in his pocket, at which point he magically teleported himself out of the maze. Knowing this, can you determine whether you will gain or lose a coin when you leave the central room through the north door? [b]p5.[/b] Bill and Charlie are playing a game on an infinite strip of graph paper. On Bill’s turn, he marks two empty squares of his choice (not necessarily adjacent) with crosses. Charlie, on his turn, can erase any number of crosses, as long as they are all adjacent to each other. Bill wants to create a line of $2013$ crosses in a row. Can Charlie stop him? [u]Round 2 [/u] [b]p6.[/b] $1000$ non-zero numbers are written around a circle and every other number is underlined. It happens that each underlined number is equal to the sum of its two neighbors and that each non-underlined number is equal to the product of its two neighbors. What could the sum of all the numbers written on the circle be? [b]p7.[/b] A grasshopper is sitting at the edge of a circle of radius $3$ inches. He can hop exactly $4$ inches in any direction, as long as he stays within the circle. Which points inside the circle can the grasshopper reach if he can make as many jumps as he likes? [img]https://cdn.artofproblemsolving.com/attachments/1/d/39b34b2b4afe607c1232f4ce9dec040a34b0c8.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $K(O,R)$ be a circle with center $O$ and radious $R$ and $(e)$ to be a line thst tangent to $K$ at $A$. A line parallel to $OA$ cuts $K$ at $B, C$, and $(e)$ at $D$, ($C$ is between $B$ and $D$). Let $E$ to be the antidiameric of $C$ with respect to $K$. $EA$ cuts $BD$ at $F$. i)Examine if $CEF$ is isosceles. ii)Prove that $2AD=EB$. iii)If $K$ si the midlpoint of $CF$, prove that $AB=KO$. iv)If $R=\frac{5}{2}, AD=\frac{3}{2}$, calculate the area of $EBF$

In a triangle $ABC$, it is known that $\angle A=100^{\circ}$ and $AB=AC$. The internal angle bisector $BD$ has length $20$ units. Find the length of $BC$ to the nearest integer, given that $\sin 10^{\circ} \approx 0.174$

A circle passes through vertex $B$ of the triangle $ABC$, intersects its sides $ AB $and $BC$ at points $K$ and $L$, respectively, and touches the side $ AC$ at its midpoint $M$. The point $N$ on the arc $BL$ (which does not contain $K$) is such that $\angle LKN = \angle ACB$. Find $\angle BAC $ given that the triangle $CKN$ is equilateral.

We consider two circles $k_1(M_1, r_1)$ and $k_2(M_2, r_2)$ with $z = M_1M_2 > r_1+r_2$ and a common outer tangent with the tangent points $P_1$ and $P2$ (that is, they lie on the same side of the connecting line $M_1M_2$). We now change the radii so that their sum is $r_1+r_2 = c$ remains constant. What set of points does the midpoint of the tangent segment $P_1P_2$ run through, when $r_1$ varies from $0$ to $c$?

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.

Given a parallelogram $ABCD$, a point M is chosen such that $\angle DAC=\angle MAC$ and $\angle CAB=\angle MAB.$ Prove $\frac{AM}{BM}=\left(\frac{AC}{BD}\right)^2$

Let there be an incircle of triangle $ABC$, and 3 circles each inscribed between incircle and angles of $ABC$. Let $r, r_1, r_2, r_3$ be radii of these circles ($r_1, r_2, r_3 < r$). Prove that $$r_1+r_2+r_3 \geq r$$

The area of a circle centered at the origin, which is inscribed in the parabola $y=x^2-25$, can be expressed as $\tfrac ab\pi$, where $a$ and $b$ are coprime positive integers. What is the value of $a+b$?

On side $AB$ of triangle $ABC$, point $D$ is selected. Circle circumscribed around triangle $BCD$, intersects side $AC$ at point $M$, and the circumcircle of triangle $ACD$ intersects the side $BC$ at point $ N$ ($M,N \ne C$). Let $O$ be the circumcenter of the triangle $CMN$. Prove that line $OD$ is perpendicular to side $AB$.

Consider a right -angle triangle $ABC$ with $A=90^{o}$, $AB=c$ and $AC=b$. Let $P\in AC$ and $Q\in AB$ such that $\angle APQ=\angle ABC$ and $\angle AQP = \angle ACB$. Calculate $PQ+PE+QF$, where $E$ and $F$ are the projections of $B$ and $Q$ onto $BC$, respectively.

Length of sides of regular hexagon $ABCDEF$ is $a$. Two moving points $M,N$ moves on sides $BC,DE$, satisfy that $\angle MAN=\frac{\pi}{3}$. Prove that $AM\cdot AN-BM\cdot DN$ is a definite value.

In a triangle $ABC$, let $BD$ be its altitude and let $H$ be its orthocenter. The perpendicular bisector of of $HD$ meets $(BCD)$ at $P, Q$. Prove that $\angle APB+\angle AQB=180^{o}$

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a function satisfying the following property: If $A, B, C \in \mathbb{R}^2$ are the vertices of an equilateral triangle with sides of length $1$, then $$f(A) + f(B) + f(C) = 0.$$ Show that $f(x) = 0$ for all $x \in \mathbb{R}^2$. [i]Proposed by Ilir Snopce[/i]

A quadrilateral $ABCD$ is such that the sides $AB$ and $DC$ are parallel, and $|BC| =|AB| + |CD|$. Prove that the angle bisectors of the angles $\angle ABC$ and $\angle BCD$ intersect at right angles on the side $AD$.

Inside a convex quadrilateral \( ABCD \), a point \( P \) is chosen such that \[ \angle PAD = \angle PAB = \angle PBC = \angle PCB = \angle PDA = 30^\circ. \] Prove that \( \angle CDP = 30^\circ \). [i]Proposed by Vadym Solomka[/i]