Let $ABC$ be a triangle, and $C^{\prime}$ and $A^{\prime}$ the midpoints of its sides $AB$ and $BC$. Consider two lines $g$ and $g^{\prime}$ which both pass through the vertex $A$ and are symmetric to each other with respect to the angle bisector of the angle $CAB$. Further, let $Y$ and $Y^{\prime}$ be the orthogonal projections of the point $B$ on these lines $g$ and $g^{\prime}$. Show that the points $Y$ and $Y^{\prime}$ are symmetric to each other with respect to the line $C^{\prime}A^{\prime}$.

Consider a $n$-sided polygon inscribed in a circle ($n \geq 4$). Partition the polygon into $n-2$ triangles using [b]non-intersecting[/b] diagnols. Prove that, irrespective of the triangulation, the sum of the in-radii of the triangles is a constant.

In an acute-angled triangle, each angle is an integral number of degrees, and the smallest angle is one-fifth of the largest one. Find these angles. (G Galperin)

Consider a regular $2n$-gon and the $n$ diagonals of it that pass through its center. Let $P$ be a point of the inscribed circle and let $a_1, a_2, \ldots , a_n$ be the angles in which the diagonals mentioned are visible from the point $P$. Prove that \[\sum_{i=1}^n \tan^2 a_i = 2n \frac{\cos^2 \frac{\pi}{2n}}{\sin^4 \frac{\pi}{2n}}.\]

Find $n$ such that $n - 76$ and $n + 76$ are both cubes of positive integers.

A point $ P$ inside a circle is such that there are three chords of the same length passing through $ P$. Prove that $ P$ is the center of the circle.

Show that for every convex polygon there is a circle passing through three consecutive vertices of the polygon and containing the entire polygon

Find the smallest number $a$ such that a square of side $a$ can contain five disks of radius $1$, so that no two of the disks have a common interior point.

A triangle $ABC$ with an obtuse angle at the vertex $C$ is inscribed in a circle with a center at point $O$. Circumcircle of triangle $AOB$ centered at point $P$ intersects line $AC$ at points $A$ and $A_1$, line $BC$ at points $B$ and $B_1$, and the perpendicular bisector of the segment $PC$ at points $D$ and $E$. Prove that points $D$ and $E$ together with the centers of the circumscribed circles of triangles $A_1OC$ and $B_1OC$ lie on one circle.

Let $H$ be the orthocenter of triangle $ABC$, $X$ be an arbitrary point. A circle with a diameter of $XH$ intersects lines $AH, BH, CH$ at points $A_1, B_1, C_1$ for the second time, and lines $AX BX, CX$ at points $A_2, B_2, C_2$. Prove that lines A$_1A_2, B_1B_2, C_1C_2$ intersect at one point.

Let $ABC$ be a triangle with orthocenter $H$. Let $\ell_b, \ell_c$ be the reflection of lines $AB$ and $AC$ about $AH$ respectively. Suppose $\ell_b$ intersect $CH$ at $P$, and $\ell_c$ intersect $BH$ at $Q$. Prove that $AH, PQ, BC$ are concurrent. [i]Proposed by Ivan Chan Kai Chin[/i]

An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the circle is $3$, and the area of the trapezoid is $72$. Let the parallel sides of the trapezoid have lengths $r$ and $s$, with $r \neq s$. Find $r^2+s^2$

Construct a quadrilateral with three sides $1$, $4$ and $3$ so that a circle could be circumscribed around it.

Prove that from any finite set of points on the plane, you can remove a point from the bottom in such a way that the remaining set can be split into two parts of smaller diameter. (Diameter is the maximum distance between points of the set.) [hide=original wording]Докажите, что из любого конечного множества точек на плоскости можно так удалитьо дну точку, что оставшееся множество можно разбить на две части меньшего диаметра. (Диаметр—это максимальное расстояние между точками множества.)[/hide]

In a parallelogram $ABCD$ , $BD$ is the largest diagonal. By matching $B$ with $D$ by a bend, a regular pentagon is formed. Calculate the measures of the angles formed by the diagonal $BD$ with each of the sides of the parallelogram.

Prove that an octagon, whose all angles are equal and all sides have rational length, has a center of symmetry.

Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$. The lines $AB$ and $CD$ meet at $P$, the lines $AD$ and $BC$ meet at $Q$, and the diagonals $AC$ and $BD$ meet at $R$. Let $M$ be the midpoint of the segment $PQ$, and let $K$ be the common point of the segment $MR$ and the circle $\omega$. Prove that the circumcircle of the triangle $KPQ$ and $\omega$ are tangent to one another.

Let $BM$ be a median of right-angled nonisosceles triangle $ABC$ ($\angle B = 90$), and $H_a$, $H_c$ be the orthocenters of triangles $ABM$, $CBM$ respectively. Lines $AH_c$ and $CH_a$ meet at point $K$. Prove that $\angle MBK = 90$.

A line meets a segment $AB$ at point $C$. Which is the maximal number of points $X$ of this line such that one of angles $AXC$ and $BXC$ is equlal to a half of the second one?

Let $ABC$ be triangle with $A<B<C$ and inscribed in a circle $(O)$. On the minor arc $ABC$ of $(O)$ and does not contain point $A$, choose an arbitrary point $D$. Suppose $CD$ meets $AB$ at $E$ and $BD$ meets $AC$ at $F$. Let $O_1$ be the incenter of triangle $EBD$ touches with $EB,ED$ and tangent to $(O)$. Let $O_2$ be the incenter of triangle $FCD$, touches with $FC,FD$ and tangent to $(O)$. a) $M$ is a tangency point of $O_1$ with $BE$ and $N$ is a tangency point of $O_2$ with $CF$. Prove that the circle with diameter $MN$ has a fixed point. b) A line through $M$ is parallel to $CE$ meets $AC$ at $P$, a line through $N$ is parallel to $BF$ meets $AB$ at $Q$. Prove that the circumcircles of triangles $(AMP),(ANQ)$ are all tangent to a fixed circle.

[u]Round 1[/u] [b]p1.[/b] Alice and Bob played $25$ games of rock-paper-scissors. Alice played rock $12$ times, scissors $6$ times, and paper $7$ times. Bob played rock $13$ times, scissors $9$ times, and paper $3$ times. If there were no ties, who won the most games? (Remember, in each game each player picks one of rock, paper, or scissors. Rock beats scissors, scissors beat paper, and paper beats rock. If they choose the same object, the result is a tie.) [b]p2.[/b] On the planet Vulcan there are eight big volcanoes and six small volcanoes. Big volcanoes erupt every three years and small volcanoes erupt every two years. In the past five years, there were $30$ eruptions. How many volcanoes could erupt this year? [b]p3.[/b] A tangle is a sequence of digits constructed by picking a number $N\ge 0$ and writing the integers from $0$ to $N$ in some order, with no spaces. For example, $010123459876$ is a tangle with $N = 10$. A palindromic sequence reads the same forward or backward, such as $878$ or $6226$. The shortest palindromic tangle is $0$. How long is the second-shortest palindromic tangle? [b]p4.[/b] Balls numbered $1$ to $N$ have been randomly arranged in a long input tube that feeds into the upper left square of an $8 \times 8$ board. An empty exit tube leads out of the lower right square of the board. Your goal is to arrange the balls in order from $1$ to $N$ in the exit tube. As a move, you may 1. move the next ball in line from the input tube into the upper left square of the board, 2. move a ball already on the board to an adjacent square to its right or below, or 3. move a ball from the lower right square into the exit tube. No square may ever hold more than one ball. What is the largest number $N$ for which you can achieve your goal, no matter how the balls are initially arranged? You can see the order of the balls in the input tube before you start. [img]https://cdn.artofproblemsolving.com/attachments/1/8/bbce92750b01052db82d58b96584a36fb5ca5b.png[/img] [b]p5.[/b] A $2018 \times 2018$ board is covered by non-overlapping $2 \times 1$ dominoes, with each domino covering two squares of the board. From a given square, a robot takes one step to the other square of the domino it is on and then takes one more step in the same direction. Could the robot continue moving this way forever without falling off the board? [img]https://cdn.artofproblemsolving.com/attachments/9/c/da86ca4ff0300eca8e625dff891ed1769d44a8.png[/img] [u]Round 2[/u] [b]p6.[/b] Seventeen teams participated in a soccer tournament where a win is worth $1$ point, a tie is worth $0$ points, and a loss is worth $-1$ point. Each team played each other team exactly once. At least $\frac34$ of all games ended in a tie. Show that there must be two teams with the same number of points at the end of the tournament. [b]p7.[/b] The city of Old Haven is known for having a large number of secret societies. Any person may be a member of multiple societies. A secret society is called influential if its membership includes at least half the population of Old Haven. Today, there are $2018$ influential secret societies. Show that it is possible to form a council of at most $11$ people such that each influential secret society has at least one member on the council. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $C$ be a circle of radius $1$ and $O$ its center. Let $\overline{AB}$ be a chord of the circle and $D$ a point on $\overline{AB}$ such that $OD =\frac{\sqrt2}{2}$ such that $D$ is closer to $ A$ than it is to $ B$, and if the perpendicular line at $D$ with respect to $\overline{AB}$ intersects the circle at $E $and $F$, $AD = DE$. The area of the region of the circle enclosed by $\overline{AD}$, $\overline{DE}$, and the minor arc $AE$ may be expressed as $\frac{a + b\sqrt{c} + d\pi}{e}$ where $a, b, c, d, e$ are integers, gcd $(a, b, d, e) = 1$, and $c$ is squarefree. Find $a + b + c + d + e$

In a non-isosceles triangle $\vartriangle ABC$ we have $\angle BAC = 60^o$. Let $D$ be the intersection of the angular bisector of $\angle BAC$ with side $BC, O$ the centre of the circumcircle of $\vartriangle ABC$ and $E$ the intersection of $AO$ and $BC$. Prove that $\angle AED + \angle ADO = 90^o$.

Consider an acute-angled triangle $\triangle ABC$ ($AC>AB$) with its orthocenter $H$ and circumcircle $\Gamma$.Points $M$,$P$ are midpoints of $BC$ and $AH$ respectively.The line $\overline{AM}$ meets $\Gamma$ again at $X$ and point $N$ lies on the line $\overline{BC}$ so that $\overline{NX}$ is tangent to $\Gamma$. Points $J$ and $K$ lie on the circle with diameter $MP$ such that $\angle AJP=\angle HNM$ ($B$ and $J$ lie one the same side of $\overline{AH}$) and circle $\omega_1$, passing through $K,H$, and $J$, and circle $\omega_2$ passing through $K,M$, and $N$, are externally tangent to each other. Prove that the common external tangents of $\omega_1$ and $\omega_2$ meet on the line $\overline{NH}$. [i]Proposed by Alireza Dadgarnia[/i]

Triangle $ABC$ is right-angled at $B$ and has incentre $I$. Points $D$, $E$ and $F$ are the points where the incircle of the triangle touches the sides $BC$, $AC$ and AB respectively. Lines $CI$ and $EF$ intersect at point $P$. Lines $DP$ and $AB$ intersect at point $Q$. Prove that $AQ = BF$.