Let $MAZN$ be an isosceles trapezium inscribed in a circle $(c)$ with centre $O$. Assume that $MN$ is a diameter of $(c)$ and let $ B$ be the midpoint of $AZ$. Let $(\epsilon)$ be the perpendicular line on $AZ$ passing through $ A$. Let $C$ be a point on $(\epsilon)$, let $E$ be the point of intersection of $CB$ with $(c)$ and assume that $AE$ is perpendicular to $CB$. Let $D$ be the point of intersection of $CZ$ with $(c)$ and let $F$ be the antidiametric point of $D$ on $(c)$. Let $ P$ be the point of intersection of $FE$ and $CZ$. Assume that the tangents of $(c)$ at the points $M$ and $Z$ meet the lines $AZ$ and $PA$ at the points $K$ and $T$ respectively. Prove that $OK$ is perpendicular to $TM$. Theoklitos Parayiou, Cyprus

Let $ABCD$ be a convex and cyclic quadrilateral. Let $AD\cap BC=\{E\}$, and let $M,N$ be points on $AD,BC$ such that $AM:MD=BN:NC$. Circle around $\triangle EMN$ intersects circle around $ABCD$ at $X,Y$ prove that $AB,CD$ and $XY$ are either parallel or concurrent.

The circles $ C_{1}$ and $ C_{2}$ touch externally at $ M$ and the radius of $ C_{2}$ is larger than that of $ C_{1}$. $ A$ is any point on $ C_{2}$ which does not lie on the line joining the centers of the circles. $ B$ and $ C$ are points on $ C_{1}$ such that $ AB$ and $ AC$ are tangent to $ C_{1}$. The lines $ BM$, $ CM$ intersect $ C_{2}$ again at $ E$, $ F$ respectively. $ D$ is the intersection of the tangent at $ A$ and the line $ EF$. Show that the locus of $ D$ as $ A$ varies is a straight line.

Let \(ABCDEF\) be a convex cyclic hexagon such that quadrilateral \(ABDF\) is a square, and the incenter of \(\triangle ACE\) lines on \(\overline{BF}\). Diagonal \(CE\) intersects diagonals \(BD\) and \(DF\) at points \(P\) and \(Q\), respectively. Prove that the circumcircle of \(\triangle DPQ\) is tangent to \(\overline{BF}\). [i]Proposed by Elliott Liu[/i]

Consider an octogon with equal angles and rational side lengths. Prove that it has a symmetry center.

In a cyclic quadrilateral $ABCD$ with circumcenter $O$, the diagonals $AC$ and $BD$ intersect at $X$. Let the circumcircles of triangles $AXD$ and $BXC$ intersect at $Y$. Let the circumcircles of triangles $AXB$ and $CXD$ intersect at $Z$. If $O$ lies inside $ABCD$ and if the points $O,X,Y,Z$ are all distinct, prove that $O,X,Y,Z$ lie on a circle.

Triangle $ABC$ is inscribed in circle $\omega$. Circle $\gamma$ is tangent to $AB$ and $AC$ at points $P$ and $Q$ respectively. Also circle $\gamma$ is tangent to circle $\omega$ at point $S$. Let the intesection of $AS$ and $PQ$ be $T$. Prove that $\angle{BTP}=\angle{CTQ}$.

A finite collection of squares has total area $4$. Show that they can be arranged to cover a square of side $1$.

Point $M{}$ is the middle of the side side $AB$ of the isosceles triangle $ABC$. On the extension of the base $AC$, point $D{}$ is marked such that $C{}$ is between $A{}$ and $D{}$, and point $E{}$ is marked on the segment $BM$. The circumcircle of the triangle $CDE$ intersects the segment $ME$ a second time at point $F$. Prove that it is possible to make a triangle from the segments $AD, DE$ and $AF$.

Find the maximum and minimum areas of the region enclosed by the curve $y=|x|e^{|x|}$ and the line $y=a\ (0\leq a\leq e)$ at $[-1,\ 1]$.

Fuzzy draws a segment of positive length in a plane. How many locations can Fuzzy place another point in the same plane to form a non-degenerate isosceles right triangle with vertices consisting of his new point and the endpoints of the segment? [i]Proposed by Timothy Qian[/i]

Let $ABC$ be a triangle and points $P,Q,R$ be on the sides $AB,BC,AC$, respectively. Now, let $A',B',C'$ be on the segments $PR,QP,RQ$ in a way that $AB||A'B'$ , $BC||B'C'$ and $AC||A'C'$. Prove that: \[ \frac{AB}{A'B'}=\frac{S_{PQR}}{S_{A'B'C'}}. \] Where $S_{XYZ}$ is the surface of the triangle $XYZ$.

Inside an $11\times 11$ square, Pablo drew a rectangle and extending its sides divided the square into $5$ rectangles, as shown in the figure. [img]https://cdn.artofproblemsolving.com/attachments/5/a/7774da7085f283b3aae74fb5ff472572571827.gif[/img] Sofía did the same, but she also managed to make the lengths of the sides of the $5$ rectangles be whole numbers between $1$ and $10$, all different. Show a figure like the one Sofia made.

The triangle $ ABC$ is given. On the extension of the side $ AB$ we construct the point $ R$ with $ BR \equal{} BC$, where $ AR > BR$ and on the extension of the side $ AC$ we construct the point $ S$ with $ CS \equal{} CB$, where $ AS > CS$. Let $ A_1$ be the point of intersection of the diagonals of the quadrilateral $ BRSC$. Analogous we construct the point $ T$ on the extension of the side $ BC$, where $ CT \equal{} CA$ and $ BT > CT$ and on the extension of the side $ BA$ we construct the point $ U$ with $ AU \equal{} AC$, where $ BU > AU$. Let $ B_1$ be the point of intersection of the diagonals of the quadrilateral $ CTUA$. Likewise we construct the point $ V$ on the extension of the side $ CA$, where $ AV \equal{} AB$ and $ CV > AV$ and on the extension of the side $ CB$ we construct the point $ W$ with $ BW \equal{} BA$ and $ CW > BW$. Let $ C_1$ be the point of intersection of the diagonals of the quadrilateral $ AVWB$. Show that the area of the hexagon $ AC_1BA_1CB_1$ is equal to the sum of the areas of the triangles $ ABC$ and $ A_1B_1C_1$.

$A, B$ are points on circle $O$ satisfying $\angle AOB < 120^{\circ} $. $C$ is a point on the tangent line of $O$ passing through $A$ satisfying $AB=AC$ and $\angle BAC < 90^{\circ} $. $D$ is the intersection of $O$ and $BC$ not $B$, and $I$ is the incenter of $ABD$. Prove that $AE=AC$ where $E$ is the intersection of $CI$ and $AD$.

In a triangle $ABC$, let $H_a$, $H_b$ and $H_c$ denote the base points of the altitudes on the sides $BC$, $CA$ and $AB$, respectively. Determine for which triangles $ABC$ two of the lengths $H_aH_b$, $H_bH_c$ and $H_aH_c$ are equal.

Given a circle (O,R). A point P lies inside the circle, OP=d, d<R. We consider quadrilaterals ABCD, inscribed in (O), such that AC is perp to BD at point P. Evaluate the maximum and minimum values of the perimeter of ABCD in terms of R and d.

In a tetrahedron, segments connecting the midpoints of heights with the orthocenters of the faces to which these heights are drawn intersect at one point. Prove that in such a tetrahedron all faces are equal or there are perpendicular edges. (Yu. Blinkov)

In tetrahedron $SABC$, the circumcircles of faces $SAB$, $SBC$, and $SCA$ each have radius $108$. The inscribed sphere of $SABC$, centered at $I$, has radius $35.$ Additionally, $SI = 125$. Let $R$ be the largest possible value of the circumradius of face $ABC$. Given that $R$ can be expressed in the form $\sqrt{\frac{m}{n}}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Author: Alex Zhu[/i]

A circle with center $O$ is inscribed in an angle. Let $A$ be the reflection of $O$ across one side of the angle. Tangents to the circle from $A$ intersect the other side of the angle at points $B$ and $C$. Prove that the circumcenter of triangle $ABC$ lies on the bisector of the original angle. (I.Sharygin)

Tim the Beaver can make three different types of geometrical figures: squares, regular hexagons, and regular octagons. Tim makes a random sequence $F_0$, $F_1$, $F_2$, $F_3$, $...$ of figures as follows: $\bullet$ $F_0$ is a square. $\bullet$ For every positive integer $i$, $F_i$ is randomly chosen to be one of the $2$ figures distinct from $F_{i-1}$ (each chosen with equal probability $\frac12$ ). $\bullet$ Tim takes $4$ seconds to make squares, $6$ to make hexagons, and $8$ to make octagons. He makes one figure after another, with no breaks in between. Suppose that exactly $17$ seconds after he starts making $F_0$, Tim is making a figure with $n$ sides. What is the expected value of $n$?

The perimeter of a triangle is a natural number, its circumradius is equal to $\frac{65}{8}$, and the inradius is equal to $4$. Find the sides of the triangle.

You are given a regular hexagon. We say that a square is inscribed in the hexagon if it can be drawn in the interior such that all the four vertices lie on the perimeter of the hexagon. [b](a)[/b] A line segment has its endpoints on opposite edges of the hexagon. Show that, it passes through the centre of the hexagon if and only if it divides the two edges in the same ratio. [b](b)[/b] Suppose, a square $ABCD$ is inscribed in the hexagon such that $A$ and $C$ are on the opposite sides of the hexagon. Prove that, centre of the square is same as that of the hexagon. [b](c)[/b] Suppose, the side of the hexagon is of length $1$. Then find the length of the side of the inscribed square whose one pair of opposite sides is parallel to a pair of opposite sides of the hexagon. [b](d)[/b] Show that, up to rotation, there is a unique way of inscribing a square in a regular hexagon.

Is it possible to draw a closed $20$-link polyline on the plane and number its links with the numbers $1, 2, 3, \ldots, 20$ in the order of traversal so that for each natural $i = 1, 2, 3, \ldots, 10$ the links numbered $i$ and $10+i$ intersect each other and do not intersect the other links? [i] I. Efremov[/i]

Let $ABC$ be an acute angled triangle satisfying the conditions $AB>BC$ and $AC>BC$. Denote by $O$ and $H$ the circumcentre and orthocentre, respectively, of the triangle $ABC.$ Suppose that the circumcircle of the triangle $AHC$ intersects the line $AB$ at $M$ different from $A$, and the circumcircle of the triangle $AHB$ intersects the line $AC$ at $N$ different from $A.$ Prove that the circumcentre of the triangle $MNH$ lies on the line $OH$.