The tangent at $C$ to $\Omega$, the circumcircle of scalene triangle $ABC$ intersects $AB$ at $D$. Through point $D$, a line is drawn that intersects segments $AC$ and $BC$ at $K$ and $L$ respectively. On the segment $AB$ points $M$ and $N$ are marked such that $AC \parallel NL$ and $BC \parallel KM$. Lines $NL$ and $KM$ intersect at point $P$ lying inside the triangle $ABC$. Let $\omega$ be the circumcircle of $MNP$. Suppose $CP$ intersects $\omega$ again at $Q$. Show that $DQ$ is tangent to $\omega$.

What is the biggest shadow that a cube of side length $1$ can have, with the sun at its peak? Note: "The biggest shadow of a figure with the sun at its peak" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane.

Three circles $S_A, S_B$, and $S_C$ in the plane with centers in $A, B$, and $C$, respectively, are mutually tangential on the outside. The touchpoint between $S_A$ and $S_B$ we call $C'$, the one $S_A$ between $S_C$ we call $B'$, and the one between $S_B$ and $S_C$ we call $A'$. The common tangent between $S_A$ and $S_C$ (passing through B') we call $\ell_B$, and the common tangent between $S_B$ and $S_C$ (passing through $A'$) we call $\ell_A$. The intersection point of $\ell_A$ and $\ell_B$ is called $X$. The point $Y$ is located so that $\angle XBY$ and $\angle YAX$ are both right angles. Show that the points $X, Y$, and $C'$ lie on a line if and only if $AC = BC$.

A circle inscribed within quadrilateral $ABCD$ is tangent to $AB$ at $E$, to $BC$ at $F$, to $CD$ at $G$, and to $DA$ at $H$. Suppose that $AE = 6$, $EB = 30$, $CG = 10$, and $GD = 2$. Compute $EF^2 + F G^2 + GH^2 + HE^2$. .

Let $P$ be a convex polyhedron. Jaroslav writes a non-negative real number to every vertex of $P$ in such a way that the sum of these numbers is $1$. Afterwards, to every edge he writes the product of the numbers at the two endpoints of that edge. Prove that the sum of the numbers at the edges is at most $\frac{3}{8}$.

Let $O$ be the circumcenter of triangle $ABC,$ and $D$ be an arbitrary point on the circumcircle of $ABC.$ Let points $X, Y$ and $Z$ be the orthogonal projections of point $D$ onto lines $OA, OB$ and $OC,$ respectively. Prove that the incenter of triangle $XYZ$ is on the Simson-Wallace line of triangle $ABC$ corresponding to point $D.$

A convex quadrilateral $ABCD$ will be called [i]rude[/i] if there exists a convex quadrilateral $PQRS$ whose points are all in the interior or on the sides of quadrilateral $ABCD$ such that the sum of diagonals of $PQRS$ is larger than the sum of diagonals of $ABCD$. Let $r>0$ be a real number. Let us assume that a convex quadrilateral $ABCD$ is not rude, but every quadrilateral $A'BCD$ such that $A'\neq A$ and $A'A\leq r$ is rude. Find all possible values of the largest angle of $ABCD$.

$ABC$ is an acute-angled triangle with circumcenter $O$. The circumcircle of $ABO$ intersects$ AC$ and $BC$ at $M$ and $N$. Show that the circumradii of $ABO$ and $MNC$ are the same.

Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is $ 4 : 3$. The horizontal length of a “$ 27$-inch” television screen is closest, in inches, to which of the following? [asy]import math; unitsize(7mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); draw((0,0)--(4,0)--(4,3)--(0,3)--(0,0)--(4,3)); fill((0,0)--(4,0)--(4,3)--cycle,mediumgray); label(rotate(aTan(3.0/4.0))*"Diagonal",(2,1.5),NW); label(rotate(90)*"Height",(4,1.5),E); label("Length",(2,0),S);[/asy]$ \textbf{(A)}\ 20 \qquad \textbf{(B)}\ 20.5 \qquad \textbf{(C)}\ 21 \qquad \textbf{(D)}\ 21.5 \qquad \textbf{(E)}\ 22$

Rectangle $ABCD$ has $AB=3$ and $BC=4.$ Point $E$ is the foot of the perpendicular from $B$ to diagonal $\overline{AC}.$ What is the area of $\triangle ADE?$ $\textbf{(A)} \text{ 1} \qquad \textbf{(B)} \text{ }\frac{42}{25} \qquad \textbf{(C)} \text{ }\frac{28}{15} \qquad \textbf{(D)} \text{ 2} \qquad \textbf{(E)} \text{ }\frac{54}{25}$

Different points $A,B,C,D,E,F$ lie on circle $k$ in this order. The tangents to $k$ in the points $A$ and $D$ and the lines $BF$ and $CE$ have a common point $P$. Prove that the lines $AD,BC$ and $EF$ also have a common point or are parallel.

Let $ABCD$ be a quadrilateral, and $P,Q,R,S$ are the midpoints of $AB, BC, CD, DA$ respectively. Let $O$ be the intersection between $PR$ and $QS$. Prove that $PO = OR$ and $QO = OS$.

We are given a convex pentagon $ABCDE$ in the coordinate plane such that $A$, $B$, $C$, $D$, $E$ are lattice points. Let $Q$ denote the convex pentagon bounded by the five diagonals of the pentagon $ABCDE$ (so that the vertices of $Q$ are the interior points of intersection of diagonals of the pentagon $ABCDE$). Prove that there exists a lattice point inside of $Q$ or on the boundary of $Q$.

Let $ABC$ be a triangle with $AB=5$, $BC=7$, and $CA=8$. Let $D$ be a point on $BC$, and define points $B'$ and $C'$ on line $AD$ (or its extension) such that $BB'\perp AD$ and $CC'\perp AD$. If $B'A=B'C'$, then the ratio $BD:DC$ can be expressed in the form $m:n$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$. [i]Proposed by Michael Ren[/i]

Anton is playing a game with shapes. He starts with a circle $\omega_1$ of radius $1$, and to get a new circle $\omega_2$, he circumscribes a square about $\omega_1$ and then circumscribes circle $\omega_2$ about that square. To get another new circle $\omega_3$, he circumscribes a regular octagon about circle $\omega_2$ and then circumscribes circle $\omega_3$ about that octagon. He continues like this, circumscribing a $2n$-gon about $\omega_{n-1}$ and then circumscribing a new circle $\omega_n$ about the $2n$-gon. As $n$ increases, the area of $\omega_n$ approaches a constant $A$. Compute $A$.

Inside the circle $ S $ there is a circle $ T $ and circles $ K_1, K_2, \ldots, K_n $ tangent externally to $ T $ and internally to $ S $, and the circle $ K_1 $ is tangent to $ K_2 $, $ K_2 $ tangent to $ K_3 $ etc. Prove that the points of tangency of the circles $ K_1 $ with $ K_2 $, $ K_2 $ with $ K_3 $ etc. lie on the circle.

Let $ABC$ be a triangle with $\angle C=90^{\circ}$, and $K, L $ be the midpoints of the minor arcs AC and BC of its circumcircle. Segment $KL$ meets $AC$,at point $N$. Find angle $NIC$ where $I$is the incenter of $ABC$.

Let $a\leqslant b\leqslant c$ be non-negative integers. A triangle on a checkered plane with vertices in the nodes of the grid is called an $(a,b,c)$[i]-triangle[/i] if there are exactly $a{}$ nodes on one side of it (not counting vertices), exactly $b{}$ nodes on the second side, and exactly $c{}$ nodes on the third side. [list] [*]Does there exist a $(9,10,11)$-triangle? [*]Find all triples of non-negative integers $a\leqslant b\leqslant c$ for which there exists an $(a,b,c)$-triangle. [*]For each such triple, find the minimum possible area of the $(a,b,c)$-triangle. [/list] [i]Proposed by P. Kozhevnikov[/i]

Convex quadrilateral $ABCD$ is cyclic in circle $(O)$, $P$ is the intersection of the diagonals $AC$ and $BD$. Circle $(O_{1})$ passes through $P$ and $B$, circle $(O_{2})$ passes through $P$ and $A$, Circles $(O_{1})$ and $(O_{2})$ intersect at $P$ and $Q$. $(O_{1})$, $(O_{2})$ intersect $(O)$ at another points $E$, $F$ (besides $B$, $A$), respectively. Prove that $PQ$, $CE$, $DF$ are concurrent or parallel.

In $\triangle ABC$, $AB = 40$, $BC = 60$, and $CA = 50$. The angle bisector of $\angle A$ intersects the circumcircle of $\triangle ABC$ at $A$ and $P$. Find $BP$. [i]Proposed by Eugene Chen[/i]

Let $I$ be the incenter of triangle $ABC$, $O_1$ a circle through $B$ tangent to $CI$, and $O_2$ a circle through $C$ tangent to $BI$. Prove that $O_1$,$O_2$ and the circumcircle of $ABC$ have a common point.

In a triangle $ABC$, points $H,M,L$ are the feet of the altitude from $C$, the median from $A$, and the angle bisector from $B$, respectively. Show that if triangle $HML$ is equilateral, then so is triangle $ABC$.

You have to divide a square paper into three parts, by two straight cuts, so that by locating these parts properly, without gaps or overlaps, an obtuse triangle is formed. Indicate how to cut the square and how to assemble the triangle with the three parts.

A set of segments with the total length less than $1$ is given on a line. Prove that every set of $n$ points on the line can be translated by a vector of length not exceeding $n/2$, so that all the obtained points are away from the given segments.

Consider a circle $C$ of center $O$ and its inner point $Q$, different from $O$. Where we must place a point $P$ on the circle $C$ so that the angle $\angle OPQ$ is the largest possible?