In a scalene triangle $ABC$, $H$ is the orthocenter, and $G$ is the centroid. Let $A_b$ and $A_c$ be points on $AB$ and $AC$, respectively, such that $B$, $C$, $A_b$, $A_c$ are cyclic, and the points $A_b$, $A_c$, $H$ are collinear. $O_a$ is the circumcenter of the triangle $AA_bA_c$. $O_b$ and $O_c$ are defined similarly. Prove that the centroid of the triangle $O_aO_bO_c$ lies on the line $HG$.

Let $\Gamma$ be a circle, $P$ be a point outside it, and $A$ and $B$ the intersection points between $\Gamma$ and the tangents from $P$ to $\Gamma$. Let $K$ be a point on the line $AB$, distinct from $A$ and $B$ and let $T$ be the second intersection point of $\Gamma$ and the circumcircle of the triangle $PBK$.Also, let $P'$ be the reflection of $P$ in point $A$. Show that $\angle PBT=\angle P'KA$

Let $M$ be a convex polygon. Externally, on its sides are built squares. It is known that the vertices of these squares, that don’t lie on $M$, lie on a circle $k$. Determine $M$ (its type).

Let $ABC$ be an acute-angled triangle with $AB> AC$ and orthocenter $H$. Let $D$ the projection of $A$ on $BC$. Let $E$ be the reflection of $C$ wrt $D$. The lines $AE$ and $BH$ intersect at point $S$. Let $N$ be the midpoint of $AE$ and let $M$ be the midpoint of $BH$. Prove that $MN$ is perpendicular to $DS$.

$ABCD$ is inscribed. Bisector of angle between diagonals intersect $AB$ anc $CD$ at $X$ and $Y$. $M,N$ are midpoints of $AD,BC$. $XM=YM$ Prove, that $XN=YN$.

Let $X = \{(x,y) \in \mathbb{R}^2 | y \geq 0, x^2+y^2 = 1\} \cup \{(x,0),-1\leq x\leq 1\} $ be the edge of the closed semicircle with radius 1. a) Let $n>1$ be an integer and $P_1,P_2,\dots,P_n \in X$. Show that there exists a permutation $\sigma \colon \{1,2,\dots,n\}\to \{1,2,\dots,n\}$ such that \[\sum_{j=1}^{n}|P_{\sigma(j+1)}-P_{\sigma(j)}|^2\leq 8\]. Where $\sigma(n+1) = \sigma(1)$. b) Find all sets $\{P_1,P_2,\dots,P_n \} \subset X$ such that for any permutation $\sigma \colon \{1,2,\dots,n\}\to \{1,2,\dots,n\}$, \[\sum_{j=1}^{n}|P_{\sigma(j+1)}-P_{\sigma(j)}|^2 \geq 8\]. Where $\sigma(n+1) = \sigma(1)$.

Let $ABC$ be a scalene triangle and $P$ be an interior point such that $\angle PBA=\angle PCA$. The lines $PB$ and $PC$ intersect the internal and external bisectors of $\angle BAC$ at $Q$ and $R$, respectively. Let $S$ be the point such that $CS$ is parallel to $AQ$ and $BS$ is parallel to $AR$. Prove that $Q$, $R$ and $S$ are colinear.

Given three equidistant parallel lines. Express by points of the corresponding lines the values of the resistance, voltage and current in a conductor so as to obtain the voltage $V = I \cdot R$ by connecting with a ruler the points denoting the resistance $R$ and the current $I$. (Each point of each scale denotes only one number). [hide=similar wording]Three parallel straight lines are given at equal distances from each other. How to depict by points of the corresponding straight lines the values of resistance, voltage and the current in the conductor, so that, applying a ruler to to points depicting the values of resistance R and values of current I, obtain on the voltage scale a point depicting the value of voltage V = I R (point each scale represents one and only one number).[/hide]

Let $A,B,P$ be points on a line $\ell$, with $P$ outside the segment $AB$. Lines $a$ and $b$ pass through $A$ and $B$ and are perpendicular to $\ell$. A line $m$ through $P$, which is neither parallel nor perpendicular to $\ell$, intersects $a$ and $b$ at $Q$ and $R$, respectively. The perpendicular from $B$ to $AR$ meets $a$ and $AR$ at $S$ and $U$, and the perpendicular from $A$ to $BQ$ meets $b$ and $BQ$ at $T$ and $V$, respectively. (a) Prove that $P,S,T$ are collinear. (b) Prove that $P,U,V$ are collinear.

Let $ABC$ be an acute triangle with circumcenter $O$, orthocenter $H$, and circumcircle $\Omega$. Let $M$ be the midpoint of $AH$ and $N$ the midpoint of $BH$. Assume the points $M$, $N$, $O$, $H$ are distinct and lie on a circle $\omega$. Prove that the circles $\omega$ and $\Omega$ are internally tangent to each other. [i]Dhroova Aiylam and Evan Chen[/i]

Let $ABCD$ be a square and $a$ be the length of his edges. The segments $AE$ and $CF$ are perpendicular on the square's plane in the same half-space and they have the length $AE=a$, $CF=b$ where $a<b<a\sqrt 3$. If $K$ denoted the set of the interior points of the square $ABCD$ determine $\min_{M\in K} \left( \max ( EM, FM ) \right) $ and $\max_{M\in K} \left( \min (EM,FM) \right)$. [i]Octavian Stanasila[/i]

[b]p1.[/b] Twelve tables are set up in a row for a Millenium party. You want to put $2000$ cupcakes on the tables so that the numbers of cupcakes on adjacent tables always differ by one (for example, if the $5$th table has $20$ cupcakes, then the $4$th table has either $19$ or $21$ cupcakes, and the $6$th table has either $19$ or $21$ cupcakes). a) Find a way to do this. b) Suppose a Y2K bug eats one of the cupcakes, so you have only $1999$ cupcakes. Show that it is impossible to arrange the cupcakes on the tables according to the above conditions. [b]p2.[/b] Let $P$ and $Q$ lie on the hypotenuse $AB$ of the right triangle $CAB$ so that $|AP|=|PQ|=|QB|=|AB|/3$. Suppose that $|CP|^2+|CQ|^2=5$. Prove that $|AB|$ has the same value for all such triangles, and find that value. Note: $|XY|$ denotes the length of the segment $XY$. [b]p3.[/b] Let $P$ be a polynomial with integer coefficients and let $a, b, c$ be integers. Suppose $P(a)=b$, $P(b)=c$, and $P(c)=a$. Prove that $a=b=c$. [b]p4.[/b] A lattice point is a point $(x,y)$ in the plane for which both $x$ and $y$ are integers. Each lattice point is painted with one of $1999$ available colors. Prove that there is a rectangle (of nonzero height and width) whose corners are lattice points of the same color. [b]p5.[/b] A $1999$-by-$1999$ chocolate bar has vertical and horizontal grooves which divide it into $1999^2$ one-by-one squares. Caesar and Brutus are playing the following game with the chocolate bar: A move consists of a player picking up one chocolate rectangle; breaking it along a groove into two smaller rectangles; and then either putting both rectangles down or eating one piece and putting the other piece down. The players move alternately. The one who cannot make a move at his turn (because there are only one-by-one squares left) loses. Caesar starts. Which player has a winning strategy? Describe a winning strategy for that player. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On a circle with center $O$ and radius $r$ are given points $A,B,C,D$ in this order such that $AB, BC$ and $CD$ have the same length $s$ and the length of $AD$ is $s+ r$.Assume that $s < r$. Determine the angles of quadrilateral $ABCD$.

Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$. Points $X$ and $Y$ lie on sides $AB$ and $CD$, respectively. Suppose the circumcircles of $CDX$ and $ABY$ meet line $XY$ again at $P$ and $Q$ respectively. Show that $OP=OQ$. [i]Proposed by Evan Chang (squareman), USA[/i]

A figure with area $1$ is cut out of paper. We divide this figure into $10$ parts and color them in $10$ different colors. Now, we turn around the piece of paper, divide the same figure on the other side of the paper in $10$ parts again (in some different way). Show that we can color these new parts in the same $10$ colors again (hereby, different parts should have different colors) such that the sum of the areas of all parts of the figure colored with the same color on both sides is $\geq \frac{1}{10}.$

In triangle $ABC$ the median $BM$ is equal to half of the side $BC$. Show that $\angle ABM = \angle BCA + \angle BAC$. [i](Proposed by Anton Trygub)[/i]

Let $ABCDEFGH$ be a cube of sidelength $a$ and such that $AG$ is one of the space diagonals. Consider paths on the surface of this cube. Then determine the set of points $P$ on the surface for which the shortest path from $P$ to $A$ and from $P$ to $G$ have the same length $l.$ Also determine all possible values of $l$ depending on $a.$

Let $k$ be the circle inscribed in convex quadrilateral $ABCD$. Lines $AD$ and $BC$ meet at $P$ ,and circumcircles of $\triangle PAB$ and $\triangle PCD$ meet in $X$ . Prove that tangents from $X$ to $k$ form equal angles with lines $AX$ and $CX$ .

Six unit disks $C_1$, $C_2$, $C_3$, $C_4$, $C_5$, $C_6$ are in the plane such that they don't intersect each other and $C_i$ is tangent to $C_{i+1}$ for $1 \le i \le 6$ (where $C_7 = C_1$). Let $C$ be the smallest circle that contains all six disks. Let $r$ be the smallest possible radius of $C$, and $R$ the largest possible radius. Find $R - r$.

Consider $4n$ points in the plane such that no three of them are collinear ($n\geq 1$). Show that the set of centroids of all the triangles formed by any three of these points contains at least $4n$ elements. [i]Radu Bumbăcea[/i]

There is a table in the shape of a $8\times 5$ rectangle with four holes on its corners. After shooting a ball from points $A, B$ and $C$ on the shown paths, will the ball fall into any of the holes after 6 reflections? (The ball reflects with the same angle after contacting the table edges.) [img]http://s5.picofile.com/file/8372960750/E01.png[/img] [i]Proposed by Hirad Alipanah[/i]

In triangle $ABC$, let $I_a$ $,I_b$, $I_c$ be the centers of the excircles tangent to sides $BC$, $CA$, $AB$, respectively. Let $P$ and $Q$ be the tangency points of the excircle of center $I_a$ with lines $AB$ and $AC$. Line $PQ$ intersects $I_aB$ and $I_aC$ at $D$ and $E$. Let $A_1$ be the intersection of $DC$ and $BE$. In an analogous way we define points $B_1$ and $C_1$. Prove that $AA_1$, $BB_1$ , $CC_1$ are concurrent.

Given right hexagon $H$ with side length $1$. On the sides of $H$ points $A_1$,$A_2$,$\ldots$,$A_k$ such that at least one of them is the midpoint of some side and for every $1 \leq i \leq k$ lines $A_{i-1}A_i$ and $A_iA_{i+1}$ form equal angles with the side, that contains the point $A_i$ (let $A_0=A_k$ and $A_{k+1}=A_1$. It is known that the length of broken line $A_1A_2\ldots A_kA_1$ is a positive integer Prove that $n$ is divisible by $3$ [i]M. Zorka[/i]

Circle $\omega$ is inscribed in quadrilateral $ABCD$ such that $AB$ and $CD$ are not parallel and intersect at point $O.$ Circle $\omega_1$ touches the side $BC$ at $K$ and touches line $AB$ and $CD$ at points which are located outside quadrilateral $ABCD;$ circle $\omega_2$ touches side $AD$ at $L$ and touches line $AB$ and $CD$ at points which are located outside quadrilateral $ABCD.$ If $O,K,$ and $L$ are collinear$,$ then show that the midpoint of side $BC,AD,$ and the center of circle $\omega$ are also collinear.

Let $ABC$ be an arbitrary acute triangle. Circle $\Gamma$ satisfies the following conditions: (i) Circle $\Gamma$ intersects all three sides of triangle $ABC.$ (ii) In the convex hexagon formed by above six intersections, the three pairs of opposite sides are parallel respectively. (The hexagon maybe degenerate, that is, two or more vertices are coincide. In this case, "opposite sides are parallel" is defined through limit opinion.) Find the locus of the center of circle $\Gamma$, and explain how to construct the locus.