For any angle α with $0 < \alpha < 180^{\circ}$, we call a closed convex planar set an $\alpha$-set if it is bounded by two circular arcs (or an arc and a line segment) whose angle of intersection is $\alpha$. Given a (closed) triangle $T$ , find the greatest $\alpha$ such that any two points in $T$ are contained in an $\alpha$-set $S \subset T .$

Let $ ABCD$ be an inscribed quadrilateral and let $ E$ be the midpoint of the diagonal $ BD$. Let $ \Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4$ be the circumcircles of triangles $ AEB$, $ BEC$, $ CED$ and $ DEA$ respectively. Prove that if $ \Gamma_4$ is tangent to the line $ CD$, then $ \Gamma_1,\Gamma_2,\Gamma_3$ are tangent to the lines $ BC,AB,AD$ respectively.

Points $A, B$ are given. Find the locus of points $C$ such that $C$, the midpoints of $AC, BC$ and the centroid of triangle $ABC$ are concyclic.

Segments $AB$ and $CD$ of length $1$ intersect at point $O$ and angle $AOC$ is equal to sixty degrees. Prove that $AC+BD \ge 1$.

Each one of 2009 distinct points in the plane is coloured in blue or red, so that on every blue-centered unit circle there are exactly two red points. Find the gratest possible number of blue points.

Find all rectangles that can be cut into $13$ equal squares.

Find the perimeter of the convex polygon whose coordinates of the vertices are the set of pairs of the integer solutions of the equation $x^2+xy = x + 2y + 9$.

Two circunmferences $ \Gamma_1$ $ \Gamma_2$ intersect at $ A$ and $ B$ $ r_1$ is the tangent from $ A$ to $ \Gamma_1$ and $ r_2$ is the tangent from $ B$ to $ \Gamma_2$ $ r_1 \cap r_2\equal{}C$ $ T\equal{} r_1 \cap \Gamma_2$ ($ T \neq A$) We consider a point $ X$ in $ \Gamma_1$ which is distinct from $ A$ and $ B$. $ XA \cap \Gamma_2 \equal{}Y$ ($ Y \neq A$) $ YB \cap XC\equal{}Z$ Prove that $ TZ \parallel XY$

A cylindrical container of revolution is partially filled with a liquid whose density we ignore. Placing it with the axis inclined $30^o$ with respect to the vertical, we observe that when removing liquid so that the level falls $1$ cm, the weight of the contents decreases $40$ g. How much will the weight of that content decrease for each centimeter that lower the level if the axis makes an angle of $45^o$ with the vertical? It is supposed that the horizontal surface of the liquid does not touch any of the bases of the container.

Let $ABC$ be a Poncelet triangle, $A_1$ is the reflection of $A$ about the incenter $I$, $A_2$ is isogonally conjugated to $A_1$ with respect to $ABC$. Find the locus of points $A_2$.

In $\triangle ABC$, point $D$ lies on side $AC$ such that $\angle ABD=\angle C$. Point $E$ lies on side $AB$ such that $BE=DE$. $M$ is the midpoint of segment $CD$. Point $H$ is the foot of the perpendicular from $A$ to $DE$. Given $AH=2-\sqrt{3}$ and $AB=1$, find the size of $\angle AME$.

Let $ABCDEFG$ be a regular heptagon with center $O$. Let $M$ be the centroid of $\triangle ABD$. Prove that $\cos^2(\angle GOM)$ is rational and determine its value. [i]Evan o'Dorney[/i]

For some values of c, the equation $x^c + y^c = z^c$ can be illustrated geometrically. For example, the case $c = 2$ can be illustrated by a right-angled triangle. By this we mean that, x, y, z is a solution of the equation $x^2 + y^2 = z^2$ if and only if there exists a right-angled triangle with catheters $x$ and $y$ and hypotenuse $z$. In this problem we will look at the cases $c = -\frac{1}{2}$ and $c = - 1$. a) Let $x, y$ and $z$ be the radii of three circles intersecting each other and a line, as shown, in the figure. Show that, $x^{-\frac{1}{2}}+ y^{-\frac{1}{2}} = z^{-\frac{1}{2}}$ [img]https://cdn.artofproblemsolving.com/attachments/5/7/5315e33e1750a3a49ae11e1b5527311117ce70.png[/img] b) Draw a geometric figure that illustrates the case in a similar way, $c = - 1$. The figure must be able to be constructed with a compass and a ruler. Describe such a construction and prove that, in the figure, lines $x, y$ and $z$ satisfy $x^{-1}+ y^{-1} = z^{-1}$. (All positive solutions of this equation should be possible values for $x, y$, and $z$ on such a figure, but you don't have to prove that.)

Points $D,E,F$ are midpoints of the sides $AB,BC,CA$ of triangle $ABC$. Angle bisectors of the angles $BDC$ and $ADC$ intersect the lines $BC$ and $AC$ respectively at the points $M$ and $N$, and the line $MN$ intersects the line $CD$ at the point $O$. Let the lines $EO$ and $FO$ intersect respectively the lines $AC$ and $BC$ at the points $P$ and $Q$. Prove that $CD=PQ$. [i](Plamen Koshlukov)[/i]

Given circle $O$ with radius $R$, the inscribed triangle $ABC$ is an acute scalene triangle, where $AB$ is the largest side. $AH_A, BH_B,CH_C$ are heights on $BC,CA,AB$. Let $D$ be the symmetric point of $H_A$ with respect to $H_BH_C$, $E$ be the symmetric point of $H_B$ with respect to $H_AH_C$. $P$ is the intersection of $AD,BE$, $H$ is the orthocentre of $\triangle ABC$. Prove: $OP\cdot OH$ is fixed, and find this value in terms of $R$. (Edited)

In tetrahedron $ABCD$, $AB=1,CD=3$, the distance between $AB$ and $CD$ is $2$, the intersection angle between $AB$ and $CD$ is $\frac{\pi}{3}$, then the volume of tetrahedron $ABCD$ is $\text{(A)}\frac{\sqrt3}{2}\qquad\text{(B)}\frac{1}{2}\qquad\text{(C)}\frac{1}{3}\qquad\text{(D)}\frac{\sqrt3}{3}$

Let $ABC$ be an acute-angled triangle having angles $\alpha,\beta,\gamma$ at vertices $A, B, C$ respectively. Let isosceles triangles $BCA_1, CAB_1, ABC_1$ be erected outwards on its sides, with apex angles $2\alpha ,2\beta ,2\gamma$ respectively. Let $A_2$ be the intersection point of lines $AA_1$ and $B_1C_1$ and let us define points $B_2$ and $C_2$ analogously. Find the exact value of the expression $$\frac{AA_1}{A_2A_1}+\frac{BB_1}{B_2B_1}+\frac{CC_1}{C_2C_1}$$

$A,B,C,D$ are points on a circle in that order. Prove that $|AB-CD|+|AD-BC| \ge 2|AC-BD|$.

Let $ABCD$ be a rectangle with: $AB=a$, $BC=b$. Inside the rectangle we have to exteriorly tangents circles such that one is tangent to the sides $AB$ and $AD$,the other is tangent to the sides $CB$ and $CD$. 1. Find the distance between the centers of the circles(using $a$ and $b$). 2. When the radiums of both circles change the tangency point between both of them changes, and describes a locus. Find that locus.

In the configuration below, $\theta$ is measured in radians, $C$ is the center of the circle, $BCD$ and $ACE$ are line segments and $AB$ is tangent to the circle at $A$. [asy] defaultpen(fontsize(10pt)+linewidth(.8pt)); pair A=(0,-1), E=(0,1), C=(0,0), D=dir(10), F=dir(190), B=(-1/sin(10*pi/180))*dir(10); fill(Arc((0,0),1,10,90)--C--D--cycle,mediumgray); fill(Arc((0,0),1,190,270)--B--F--cycle,mediumgray); draw(unitcircle); draw(A--B--D^^A--E); label("$A$",A,S); label("$B$",B,W); label("$C$",C,SE); label("$\theta$",C,SW); label("$D$",D,NE); label("$E$",E,N); [/asy] A necessary and sufficient condition for the equality of the two shaded areas, given $0 < \theta < \frac{\pi}{2}$, is $ \textbf{(A)}\ \tan \theta = \theta\qquad\textbf{(B)}\ \tan \theta = 2\theta\qquad\textbf{(C)}\ \tan \theta = 4\theta\qquad\textbf{(D)}\ \tan 2\theta = \theta\qquad \\ \textbf{(E)}\ \tan \frac{\theta}{2} = \theta$

Let $I$, $I_{a}$ be the incenter and the $A$-excenter of a triangle $ABC$; $E$, $F$ be the touching points of the incircle with $AC$, $AB$ respectively; $G$ be the common point of $BE$ and $CF$. The perpendicular to $BC$ from $G$ meets $AI$ at point $J$. Prove that $E$, $F$, $J$, $I_{a}$ are concyclic. Proposed by:Y.Shcherbatov

[u]Round 5[/u] [b]p13.[/b] Given a (not necessarily simplified) fraction $\frac{m}{n}$ , where $m, n > 6$ are positive integers, when $6$ is subtracted from both the numerator and denominator, the resulting fraction is equal to $\frac45$ of the original fraction. How many possible ordered pairs $(m, n)$ are there? [b]p14.[/b] Jamesu's favorite anime show has $3$ seasons, with $12$ episodes each. For $8$ days, Jamesu does the following: on the $n^{th}$ day, he chooses $n$ consecutive episodes of exactly one season, and watches them in order. How many ways are there for Jamesu to finish all $3$ seasons by the end of these $8$ days? (For example, on the first day, he could watch episode $5$ of the first season; on the second day, he could watch episodes $11$ and $12$ of the third season, etc.) [b]p15.[/b] Let $O$ be the center of regular octagon $ABCDEFGH$ with side length $6$. Let the altitude from $O$ meet side $AB$ at $M$, and let $BH$ meet $OM$ at $K$. Find the value of $BH \cdot BK$. [u]Round 6[/u] [b]p16.[/b] Fhomas writes the ordered pair $(2, 4)$ on a chalkboard. Every minute, he erases the two numbers $(a, b)$, and replaces them with the pair $(a^2 + b^2, 2ab)$. What is the largest number on the board after $10$ minutes have passed? [b]p17.[/b] Triangle $BAC$ has a right angle at $A$. Point $M$ is the midpoint of $BC$, and $P$ is the midpoint of $BM$. Point $D$ is the point where the angle bisector of $\angle BAC$ meets $BC$. If $\angle BPA = 90^o$, what is $\frac{PD}{DM}$? [b]p18.[/b] A square is called legendary if there exist two different positive integers $a, b$ such that the square can be tiled by an equal number of non-overlapping $a$ by $a$ squares and $b$ by $b$ squares. What is the smallest positive integer $n$ such that an $n$ by $n$ square is legendary? [u]Round 7[/u] [b]p19.[/b] Let $S(n)$ be the sum of the digits of a positive integer $n$. Let $a_1 = 2019!$, and $a_n = S(a_{n-1})$. Given that $a_3$ is even, find the smallest integer $n \ge 2$ such that $a_n = an_1$. [b]p20.[/b] The local EMCC bakery sells one cookie for $p$ dollars ($p$ is not necessarily an integer), but has a special offer, where any non-zero purchase of cookies will come with one additional free cookie. With $\$27:50$, Max is able to buy a whole number of cookies (including the free cookie) with a single purchase and no change leftover. If the price of each cookie were $3$ dollars lower, however, he would be able to buy double the number of cookies as before in a single purchase (again counting the free cookie) with no change leftover. What is the value of $p$? [b]p21.[/b] Let circle $\omega$ be inscribed in rhombus $ABCD$, with $\angle ABC < 90^o$. Let the midpoint of side $AB$ be labeled $M$, and let $\omega$ be tangent to side $AB$ at $E$. Let the line tangent to $\omega$ passing through $M$ other than line $AB$ intersect segment $BC$ at $F$. If $AE = 3$ and $BE = 12$, what is the area of $\vartriangle MFB$? [u]Round 8[/u] [b]p22.[/b] Find the remainder when $1010 \cdot 1009! + 1011 \cdot 1008! + ... + 2018 \cdot 1!$ is divided by $2019$. [b]p23.[/b] Two circles $\omega_1$ and $\omega_2$ have radii $1$ and $2$, respectively and are externally tangent to one another. Circle $\omega_3$ is externally tangent to both $\omega_1$ and $\omega_2$. Let $M$ be the common external tangent of $\omega_1$ and $\omega_3$ that doesn't intersect $\omega_2$. Similarly, let $N$ be the common external tangent of $\omega_2$ and $\omega_3$ that doesn't intersect $\omega_1$. Given that $M$ and N are parallel, find the radius of $\omega_3$. [b]p24.[/b] Mana is standing in the plane at $(0, 0)$, and wants to go to the EMCCiffel Tower at $(6, 6)$. At any point in time, Mana can attempt to move $1$ unit to an adjacent lattice point, or to make a knight's move, moving diagonally to a lattice point $\sqrt5$ units away. However, Mana is deathly afraid of negative numbers, so she will make sure never to decrease her $x$ or $y$ values. How many distinct paths can Mana take to her destination? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2949411p26408196]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In convex quadrilateral $KLMN$ side $\overline{MN}$ is perpendicular to diagonal $\overline{KM}$, side $\overline{KL}$ is perpendicular to diagonal $\overline{LN}$, $MN = 65$, and $KL = 28$. The line through $L$ perpendicular to side $\overline{KN}$ intersects diagonal $\overline{KM}$ at $O$ with $KO = 8$. Find $MO$.

Triangle $ABC$ is given with its centroid $G$ and cicumcentre $O$ is such that $GO$ is perpendicular to $AG$. Let $A'$ be the second intersection of $AG$ with circumcircle of triangle $ABC$. Let $D$ be the intersection of lines $CA'$ and $AB$ and $E$ the intersection of lines $BA'$ and $AC$. Prove that the circumcentre of triangle $ADE$ is on the circumcircle of triangle $ABC$.

A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly 1/1985. [asy] size(200); pair A=(0,1), B=(1,1), C=(1,0), D=origin; draw(A--B--C--D--A--(1,1/6)); draw(C--(0,5/6)^^B--(1/6,0)^^D--(5/6,1)); pair point=( 0.5 , 0.5 ); //label("$A$", A, dir(point--A)); //label("$B$", B, dir(point--B)); //label("$C$", C, dir(point--C)); //label("$D$", D, dir(point--D)); label("$1/n$", (11/12,1), N, fontsize(9));[/asy]