Let $ABC$ be a triangle with all angles $\leq 120^{\circ}$. Let $F$ be the Fermat point of triangle $ABC$, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of the three triangles $BFC$, $CFA$ and $AFB$, draw its Euler line - that is, the line connecting its circumcenter and its centroid. Prove that these three Euler lines pass through one common point. [i]Remark.[/i] The Fermat point $F$ is also known as the [b]first Fermat point[/b] or the [b]first Toricelli point[/b] of triangle $ABC$. [i]Floor van Lamoen[/i]

Let $ d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $ n$ vertices (where $ n>3$). Let $ p$ be its perimeter. Prove that: \[ n\minus{}3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n\plus{}1\over 2}\Bigr]\minus{}2,\] where $ [x]$ denotes the greatest integer not exceeding $ x$.

Let $ABCD$ be a convex quadrilateral whose diagonals $AC$ and $BD$ intersect in a point $P$. Prove that \[\frac{AP}{PC}=\frac{\cot \angle BAC + \cot \angle DAC}{\cot \angle BCA + \cot \angle DCA}\]

Let $ ABCD $ be an inscriptible quadrilateral and $ M $ be a point on its circumcircle, distinct from its vertices. Let $ H_1,H_2,H_3,H_4 $ be the orthocenters of $ MAB,MBC, MCD, $ respectively, $ MDA, $ and $ E,F, $ the midpoints of the segments $ AB, $ respectivley, $ CD. $ Prove that: [b]a)[/b] $ H_1H_2H_3H_4 $ is a parallelogram. [b]b)[/b] $ H_1H_3=2\cdot EF. $

An acute-angle non-isosceles triangle $ ABC $ is given. The point $ H $ is its orthocenter, the points $ O $ and $ I $ are the centers of its circumscribed and inscribed circles, respectively. The circumcircle of the triangle $ OIH $ passes through the vertex $ A $. Prove that one of the angles of the triangle is $ 60^\circ $.

Let $C_{1}$ and $C_{2}$ be concentric circles, with $C_{2}$ in the interior of $C_{1}$. From a point $A$ on $C_{1}$, draw the tangent $AB$ to $C_{2}$ $(B \in C_{2})$. Let $C$ be the second point of intersection of $AB$ and $C_{1}$,and let $D$ be the midpoint of $AB$. A line passing through $A$ intersects $C_{2}$ at $E$ and $F$ in such a way that the perpendicular bisectors of $DE$ and $CF$ intersect at a point $M$ on $AB$. Find, with proof, the ratio $AM/MC$. This question is taken from Mathematical Olympiad Challenges , the 9-th exercise in 1.3 Power of a Point.

All squares of a $ 20\times 20$ table are empty. Misha* and Sasha** in turn put chips in free squares (Misha* begins). The player after whose move there are four chips on the intersection of two rows and two columns wins. Which of the players has a winning strategy? [i]Proposed by A. Golovanov[/i] [b]US Name Conversions: [/b] [i]Misha*: Naoki Sasha**: Richard[/i]

Consider a circle with center $O$ and radius $R,$ and let $A$ and $B$ be two points in the plane of this circle. [b]a.)[/b] Draw a chord $CD$ of the circle such that $CD$ is parallel to $AB,$ and the point of the intersection $P$ of the lines $AC$ and $BD$ lies on the circle. [b]b.)[/b] Show that generally, one gets two possible points $P$ ($P_{1}$ and $P_{2}$) satisfying the condition of the above problem, and compute the distance between these two points, if the lengths $OA=a,$ $OB=b$ and $AB=d$ are given.

Let $ABCDEF$ be a convex hexagon satisfying $AC=DF$, $CE=FB$ and $EA=BD$. Prove that the lines connecting the midpoints of opposite sides of the hexagon $ABCDEF$ intersect in one point.

Equiangular hexagon $ ABCDEF$ has side lengths $ AB \equal{} CD \equal{} EF \equal{} 1$ and $ BC \equal{} DE \equal{} FA \equal{} r$. The area of $ \triangle ACE$ is $70\%$ of the area of the hexagon. What is the sum of all possible values of $ r$? $ \textbf{(A)}\ \frac {4\sqrt {3}}{3} \qquad \textbf{(B)}\ \frac {10}{3} \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ \frac {17}{4} \qquad \textbf{(E)}\ 6$

[u]Round 1[/u] [b]p1.[/b] Goldilocks enters the home of the three bears – Papa Bear, Mama Bear, and Baby Bear. Each bear is wearing a different-colored shirt – red, green, or blue. All the bears look the same to Goldilocks, so she cannot otherwise tell them apart. The bears in the red and blue shirts each make one true statement and one false statement. The bear in the red shirt says: “I'm Blue's dad. I'm Green's daughter.” The bear in the blue shirt says: “Red and Green are of opposite gender. Red and Green are my parents.” Help Goldilocks find out which bear is wearing which shirt. [b]p2.[/b] The University of Washington is holding a talent competition. The competition has five contests: math, physics, chemistry, biology, and ballroom dancing. Any student can enter into any number of the contests but only once for each one. For example, a student may participate in math, biology, and ballroom. It turned out that each student participated in an odd number of contests. Also, each contest had an odd number of participants. Was the total number of contestants odd or even? [b]p3.[/b] The $99$ greatest scientists of Mars and Venus are seated evenly around a circular table. If any scientist sees two colleagues from her own planet sitting an equal number of seats to her left and right, she waves to them. For example, if you are from Mars and the scientists sitting two seats to your left and right are also from Mars, you will wave to them. Prove that at least one of the $99$ scientists will be waving, no matter how they are seated around the table. [b]p4.[/b] One hundred boys participated in a tennis tournament in which every player played each other player exactly once and there were no ties. Prove that after the tournament, it is possible for the boys to line up for pizza so that each boy defeated the boy standing right behind him in line. [b]p5.[/b] To celebrate space exploration, the Science Fiction Museum is going to read Star Wars and Star Trek stories for $24$ hours straight. A different story will be read each hour for a total of $12$ Star Wars stories and $12$ Star Trek stories. George and Gene want to listen to exactly $6$ Star Wars and $6$ Star Trek stories. Show that no matter how the readings are scheduled, the friends can find a block of $12$ consecutive hours to listen to the stories together. [u]Round 2[/u] [b]p6.[/b] $2013$ people attended Cinderella's ball. Some of the guests were friends with each other. At midnight, the guests started turning into mice. After the first minute, everyone who had no friends at the ball turned into a mouse. After the second minute, everyone who had exactly one friend among the remaining people turned into a mouse. After the third minute, everyone who had two human friends left in the room turned into a mouse, and so on. What is the maximal number of people that could have been left at the ball after $2013$ minutes? [b]p7.[/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? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $M$ be the point of intersection of the diagonals $AC$ and $BD$ of the convex quadrilateral $ABCD$. Let$ K$ be the intersection point of the extension of side $AB$ (from $A$) with the bisector of the $\angle ACD$. If $MA \cdot MC + MA \cdot CD = MB\cdot MD$ , prove that $\angle BKC = \angle CDB$.

$ABCDEF$ is a cyclic hexagon with circumcenter $O$, and $AD, BE, CF$ are concurrent at $X$. $P$ is a point on the plane. The circumenter of $PAB$ is $O_{AB}$. Define $O_{BC}, O_{CD}$, $O_{DE}, O_{EF}, O_{FA}$ similarly. Prove that $O_{AB} O_{DE}, O_{BC}O_{EF}, O_{CD}O_{FA}$, $OX$ are concurrent.

$PQR$ is an acute scalene triangle. The altitude $PL$ and the bisector $RK$ of $\angle QRP$ meet at $H$ ($L$ on $QR$ and $K$ on $PQ$). $KM$ is the altitude of triangle $PKR$; it meets $PL$ at $N$. The circumcircle of $\triangle NKR$ meets $QR$ at $S$ other than $Q$. Prove that $SHK$ is an isosceles triangle.

Prove that a line that divides a triangle into two polygons of equal area and equal perimeter passes through the center of the circle inscribed in the triangle. Prove an analogous property for a polygon that has an inscribed circle.

Let $\mathcal{S}$ be the set of points $(x, y)$ in the Cartesian coordinate plane such that $xy > 0$ and $x^2 + y^2 + 2x + 4y \leq 2021.$ The total area of $\mathcal{S}$ can be written in the form $a\pi + b,$ where $a$ and $b$ are integers. Compute $a + b.$

Let $ABC$ be a triangle with circumscribed and inscribed circles $(O)$ and $(I)$ respectively. $AA'$,$BB'$,$CC'$ are the bisectors of triangle $ABC$. Prove that $OI$ passes through the the isogonal conjugate of point $I$ with respect to triangle $A'B'C'$.

In the rectangle $ABCD$, the ratio of the lengths of sides $BC$ and $AB$ is equal to $\sqrt{2}$. Point $X$ is marked inside this rectangle so that $AB=BX=XD$. Determine the measure of angle $BXD$.

$10$ rectangles have their vertices lie on a circle. The vertices divide the circle into $40$ equal arcs. Prove that two of the rectangles are congruent. [i]Proposed by Evan Chang (squareman), USA[/i]

Let $ABC$ be an arbitrary triangle. On the perpendicular bisector of $AB$, there is a point $P$ inside of triangle $ABC$. On the sides $BC$ and $CA$, triangles $BQC$ and $CRA$ are placed externally. These triangles satisfy $\vartriangle BPA \sim \vartriangle BQC \sim \vartriangle CRA$. (So $Q$ and $A$ lie on opposite sides of $BC$, and $R$ and $B$ lie on opposite sides of $AC$.) Show that the points $P, Q, C$ and $R$ form a parallelogram.

Let $ABC$ be an acute triangle with $AB < AC$, let $\Gamma$ be its circumcircle and let $D$ be the foot of the altitude from $A$ to $BC$. Take a point $E$ on the segment $BC$ such that $CE=BD$. Let $P$ be the point on $\Gamma$ diametrically opposite to vertex $A$. Prove that $PE$ is perpendicular to $BC$.

A nondegenerate rhombus has side length $l$, and its area is twice that of its inscribed circle. Find the radius of the inscribed circle.

A cone with semivertical angle $30^{\circ}$ is half filled with water. What is the angle it must be tilted by so that water starts spilling?

A line is drawn through a vertex of a triangle and cuts two of its middle lines (i.e. lines connecting the midpoints of two sides) in the same ratio. Determine this ratio.

Let $a,b,c$ be three positive real numbers with $a+b+c=3$. Prove that \[ (3-2a)(3-2b)(3-2c) \leq a^2b^2c^2 . \] [i]Robert Szasz[/i]