Let $ABC$ be an acute-angled triangle with $AB < AC$, and let points $D$ and $E$ be chosen on the side $AC$ and $BC$ respectively in such a way that $AD = AE = AB$. The circumcircle of $ABE$ intersects the line $AC$ at $A$ and $F$ and the line $DE$ at $E$ and $P$. Prove that $P$ is the circumcentre of $BDF$.

Let $S$ be a set of 1980 points in the plane such that the distance between every pair of them is at least 1. Prove that $S$ has a subset of 220 points such that the distance between every pair of them is at least $\sqrt{3}.$

Point $E$ is the midpoint of side $\overline{CD}$ in square $ABCD,$ and $\overline{BE}$ meets diagonal $\overline{AC}$ at $F.$ The area of quadrilateral $AFED$ is $45.$ What is the area of $ABCD?$ [asy] size(5cm); draw((0,0)--(6,0)--(6,6)--(0,6)--cycle); draw((0,6)--(6,0)); draw((3,0)--(6,6)); label("$A$",(0,6),NW); label("$B$",(6,6),NE); label("$C$",(6,0),SE); label("$D$",(0,0),SW); label("$E$",(3,0),S); label("$F$",(4,2),E); [/asy] $\textbf{(A) } 100 \qquad \textbf{(B) } 108 \qquad \textbf{(C) } 120 \qquad \textbf{(D) } 135 \qquad \textbf{(E) } 144$

Find the greatest real $k$ such that, for every tetrahedron $ABCD$ of volume $V$, the product of areas of faces $ABC,ABD$ and $ACD$ is at least $kV^2$.

$(SWE 3)$ Find the natural number $n$ with the following properties: $(1)$ Let $S = \{P_1, P_2, \cdots\}$ be an arbitrary finite set of points in the plane, and $r_j$ the distance from $P_j$ to the origin $O.$ We assign to each $P_j$ the closed disk $D_j$ with center $P_j$ and radius $r_j$. Then some $n$ of these disks contain all points of $S.$ $(2)$ $n$ is the smallest integer with the above property.

Let $ABC$ be a triangle with altitudes $AD, BE, CF$. Choose the points $A_1, B_1, C_1$ on the lines $AD, BE, CF$ respectively, such that $$\frac{AA_1}{AD}= \frac{BB_1}{BE}= \frac{CC_1}{CF} = k.$$ Find all values of $k$ such that the triangle $A_1B_1C_1$ is similar to the triangle $ABC$ for all triangles $ABC$.

For a triangle $ ABC $ which $ \angle B \ne 90^{\circ} $ and $ AB \ne AC $, define $ P_{ABC} $ as follows ; Let $ I $ be the incenter of triangle $ABC$, and let $ D, E, F $ be the intersection points with the incircle and segments $ BC, CA, AB $. Two lines $ AB $ and $ DI $ meet at $ S $ and let $ T $ be the intersection point of line $ DE $ and the line which is perpendicular with $ DF $ at $ F $. The line $ ST $ intersects line $ EF $ at $ R$. Now define $ P_{ABC} $ be one of the intersection points of the incircle and the circle with diameter $ IR $, which is located in other side with $ A $ about $ IR $. Now think of an isosceles triangle $ XYZ $ such that $ XZ = YZ > XY $. Let $ W $ be the point on the side $ YZ $ such that $ WY < XY $ and Let $ K = P_{YXW} $ and $ L = P_{ZXW} $. Prove that $ 2 KL \le XY $.

A $ 3 \times 3 \times 3$ cube is formed by gluing together 27 standard cubical dice. (On a standard die, the sum of the numbers on any pair of opposite faces is 7.) The smallest possible sum of all the numbers showing on the surface of the $ 3 \times 3 \times 3$ cube is $ \text{(A)}\ 60 \qquad \text{(B)}\ 72 \qquad \text{(C)}\ 84 \qquad \text{(D)}\ 90 \qquad \text{(E)}\ 96$

Let $S$ be the set of points inside a given equilateral triangle $ABC$ with side $1$ or on its boundary. For any $M \in S, a_M, b_M, c_M$ denote the distances from $M$ to $BC,CA,AB$, respectively. Define \[f(M) = a_M^3 (b_M - c_M) + b_M^3(c_M - a_M) + c_M^3(a_M - b_M).\] [b](a)[/b] Describe the set $\{M \in S | f(M) \geq 0\}$ geometrically. [b](b)[/b] Find the minimum and maximum values of $f(M)$ as well as the points in which these are attained.

Let $\omega$ be the circumcircle of isosceles triangle $ABC$ ($AB=AC$). Points $P$ and $Q$ lie on $\omega$ and $BC$ respectively such that $AP=AQ$ .$AP$ and $BC$ intersect at $R$. Prove that the tangents from $B$ and $C$ to the incircle of $\triangle AQR$ (different from $BC$) are concurrent on $\omega$. [i]Proposed by Ali Zamani, Hooman Fattahi[/i]

Given a circle $\Gamma$ and the positive numbers $h$ and $m$, construct with straight edge and compass a trapezoid inscribed in $\Gamma$, such that it has altitude $h$ and the sum of its parallel sides is $m$.

Let $A, B, C, D$ be points that lie on the same circle . Let $F$ be such that the arc $AF$ is congruent with the arc $BF$. Let $P$ be the intersection point of the segments $DF$ and $AC$. Let $Q$ be intersection point of the $CF$ and $BD$ segments. Prove that $PQ \parallel AB$.

Square $ABCD$ in the coordinate plane has vertices at the points $A(1,1), B(-1,1), C(-1,-1),$ and $D(1,-1).$ Consider the following four transformations: [list=] [*]$L,$ a rotation of $90^{\circ}$ counterclockwise around the origin; [*]$R,$ a rotation of $90^{\circ}$ clockwise around the origin; [*]$H,$ a reflection across the $x$-axis; and [*]$V,$ a reflection across the $y$-axis. [/list] Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying $R$ and then $V$ would send the vertex $A$ at $(1,1)$ to $(-1,-1)$ and would send the vertex $B$ at $(-1,1)$ to itself. How many sequences of $20$ transformations chosen from $\{L, R, H, V\}$ will send all of the labeled vertices back to their original positions? (For example, $R, R, V, H$ is one sequence of $4$ transformations that will send the vertices back to their original positions.) $\textbf{(A)}\ 2^{37} \qquad\textbf{(B)}\ 3\cdot 2^{36} \qquad\textbf{(C)}\ 2^{38} \qquad\textbf{(D)}\ 3\cdot 2^{37} \qquad\textbf{(E)}\ 2^{39}$

The vertices of a triangle $DEF$ lie on different sides of a triangle $ABC$. The lengths of the tangents from the incenter of $DEF$ to the excircles of $ABC$ are equal. Prove that $4S_{DEF} \ge S_{ABC}$. [i]Note: By $S_{XYZ}$ we denote the area of triangle $XYZ$.[/i]

The line $\ell$ is perpendicular to the side $AC$ of the acute triangle $ABC$ and intersects this side at point $K$, and the circumcribed circle $\vartriangle ABC$ at points $P$ and $T$ (point P on the other side of line $AC$, as the vertex $B$). Denote by $P_1$ and $T_1$ - the projections of the points $P$ and $T$ on line $AB$, with the vertices $A, B$ belong to the segment $P_1T_1$. Prove that the center of the circumscribed circle of the $\vartriangle P_1KT_1$ lies on a line containing the midline $\vartriangle ABC$, which is parallel to the side $AC$. (Anton Trygub)

The diagonals of a pentagon $ABCDE$ determine another pentagon $MNPQR$. If $MNPQR$ and $ABCDE$ are similar, must $ABCDE$ be regular?

$A,B,C$ and $D$ are four points in that order on the circumference of a circle $K$. $AB$ is perpendicular to $BC$ and $BC$ is perpendicular to $CD$. $X$ is a point on the circumference of the circle between $A$ and $D$. $AX$ extended meets $CD$ extended at $E$ and $DX$ extended meets $BA$ extended at $F$. Prove that the circumcircle of triangle $AXF$ is tangent to the circumcircle of triangle $DXE$ and that the common tangent line passes through the center of the circle $K$.

In an acute triangle $ABC$, points $D,E,F$ are the feet of the altitudes from $A,B,C$, respectively. A line through $D$ parallel to $EF$ meets $AC$ at $Q$ and $AB$ at $R$. Lines $BC$ and $EF$ intersect at $P$. Prove that the circumcircle of triangle $PQR$ passes through the midpoint of $BC$.

Given a family $C$ of circles of the same radius $R$, which completely covers the plane (that is, every point in the plane belongs to at least one circle of the family), prove that there exist two circles of the family such that the distance between their centers is less than or equal to $R\sqrt3$ .

In an acute-angled triangle $ABC$ on the sides $AB$, $BC$, $AC$ the points $H$, $L$, $K$ so that $CH \perp AB$, $HL \parallel AC$, $HK \parallel BC$. Let $P$ and $Q$ feet of altitudes of a triangle $HBL$, drawn from the vertices $H$ and $B$ respectively. Prove that the feet of the altitudes of the triangle $AKH$, drawn from the vertices $A$ and $H$ lie on the line $PQ$.

$H$ is the orthocenter of $\triangle ABC$,the circle with center $H$ passes through $A$,and it intersects with $AC,AB$ at two other points $D,E$.The orthocenter of $\triangle ADE$ is $H'$,line $AH'$ intersects with $DE$ at point $F$.Point $P$ is inside the quadrilateral $BCDE$,so that $\triangle PDE\sim\triangle PBC$.Let point $K$ be the intersection of line $HH'$ and line $PF$.Prove that $A,H,P,K$ lie on one circle. [img]https://i.ibb.co/mcyhxRM/graph.jpg[/img]

In a polygon, every external angle is one sixth of its corresponding internal angle. How many sides does the polygon have?

Does there exist a convex pentagon such that for any of its inner angles, the angle bisector contains one of the diagonals?

Let the sides of two rectangles be $ \{a,b\}$ and $ \{c,d\},$ respectively, with $ a < c \leq d < b$ and $ ab < cd.$ Prove that the first rectangle can be placed within the second one if and only if \[ \left(b^2 \minus{} a^2\right)^2 \leq \left(bc \minus{} ad \right)^2 \plus{} \left(bd \minus{} ac \right)^2.\]

A point $P$ is chosen uniformly at random in the interior of triangle $ABC$ with side lengths $AB = 5$, $BC = 12$, $CA = 13$. The probability that a circle with radius $\frac13$ centered at $P$ does not intersect the perimeter of $ABC$ can be written as $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Find $m + n$.