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 $\sqrt{3}.$

In triangle $ABC,\angle BAC = 60^o, AB = 3a$ and $AC = 4a, (a > 0)$. Let $M$ be point on the segment $AB$ such that $AM =\frac13 AB, N$ be point on the side $AC$ such that $AN =\frac12AC$. Let $I$ be midpoint of $MN$. Determine the length of $BI$. A. $\frac{a\sqrt2}{19}$ B. $\frac{2a}{\sqrt{19}}$ C. $\frac{19a\sqrt{19}}{2}$ D. $\frac{19a}{\sqrt2}$ E. $\frac{a\sqrt{19}}{2}$

Let $ABC$ be a triangle and $I$ is your incenter, let $P$ be a point in $AC$ such that $PI$ is perpendicular to $AC$, and let $D$ be the reflection of $B$ wrt circumcenter of $\triangle ABC$. The line $DI$ intersects again the circumcircle of $\triangle ABC$ in the point $Q$. Prove that $QP$ is the angle bisector of the angle $\angle AQC$.

On $\triangle ABC$ let points $D,E$ on sides $AB,BC$ respectivily such that $AD=DE=EC$ and $AE \ne DC$. Let $P$ the intersection of lines $AE, DC$, show that $\angle ABC=60$ if $AP=CP$.

A $10 \times 10 \times 10$ cube is made up up from $500$ white unit cubes and $500$ black unit cubes, arranged in such a way that every two unit cubes that shares a face are in different colors. A line is a $1 \times 1 \times 10$ portion of the cube that is parallel to one of cube’s edges. From the initial cube have been removed $100$ unit cubes such that $300$ lines of the cube has exactly one missing cube. Determine if it is possible that the number of removed black unit cubes is divisible by $4$.

Prove that every polyhedron has at least two faces with the same number of sides.

On a $2004 \times 2004$ chess table there are 2004 queens such that no two are attacking each other\footnote[1]{two queens attack each other if they lie on the same row, column or direction parallel with on of the main diagonals of the table}. Prove that there exist two queens such that in the rectangle in which the center of the squares on which the queens lie are two opposite corners, has a semiperimeter of 2004.

How many different isosceles triangles have integer side lengths and perimeter 23? $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 11$

The circle circumscribed about an acute triangle $ABC$ and the vertex $C$ are fixed. Orthocenter $H$ moves in a circle with center at point $C$. Find the locus of the midpoints of the segments connecting the feet of altitudes drawn from vertices $A$ and $B$.

The bottom, side, and front areas of a rectangular box are known. The product of these areas is equal to: $ \textbf{(A)}\ \text{the volume of the box} \qquad\textbf{(B)}\ \text{the square root of the volume} \qquad\textbf{(C)}\ \text{twice the volume}$ $ \textbf{(D)}\ \text{the square of the volume} \qquad\textbf{(E)}\ \text{the cube of the volume}$

Let $ABCD$ be a rectangle with diagonals of length $10.$ Let $P$ be the midpoint of $\overline{AD},$ $S$ be the midpoint of $\overline{BC},$ and $T$ be the midpoint of $\overline{CD}.$ Points $Q$ and $R$ are chosen on $\overline{AB}$ such that $AP=AQ$ and $BR=BS,$ and minor arcs $\widehat{PQ}$ and $\widehat{RS}$ centered at $A$ and $B,$ respectively, are drawn. Circle $\omega$ is tangent to $\overline{CD}$ at $T$ and externally tangent to $\widehat{PQ}$ and $\widehat{RS}.$ Suppose that the radius of $\omega$ is $\tfrac{43}{18}.$ Then the sum of all possible values of the area of $ABCD$ can be written in the form $\tfrac{a+b\sqrt{c}}{d},$ where $a,\ b,\ c,$ and $d$ are positive integers, $b$ and $d$ are relatively prime, and $c$ is prime. Find $a+b+c+d.$

Let $ABC$ be acute-angled triangle with circumcircle $\Gamma$. Points $H$ and $M$ are the orthocenter and the midpoint of $BC$ respectively. The line $HM$ meets the circumcircle $\omega$ of triangle $BHC$ at point $N\not= H$. Point $P$ lies on the arc $BC$ of $\omega$ not containing $H$ in such a way that $\angle HMP = 90^\circ$. The segment $PM$ meets $\Gamma$ at point $Q$. Points $B'$ and $C'$ are the reflections of $A$ about $B$ and $C$ respectively. Prove that the circumcircles of triangles $AB'C'$ and $PQN$ are tangent.

Call a right triangle $ABC$ [i]special [/i] if the lengths of its sides $AB, BC$ and$ CA$ are integers, and on each of these sides has some point $X$ (different from the vertices of $ \vartriangle ABC$), for which the lengths of the segments $AX, BX$ and $CX$ are integers numbers. Find at least one special triangle. (Maria Rozhkova)

The product of $\sqrt[3]{4}$ and $\sqrt[4]{8}$ equals $\textbf{(A) }\sqrt[7]{12}\qquad\textbf{(B) }2\sqrt[7]{12}\qquad\textbf{(C) }\sqrt[7]{32}\qquad\textbf{(D) }\sqrt[12]{32}\qquad\textbf{(E) }2\sqrt[12]{32}$

Prove that the number $1\underbrace{\hbox{0...0}}_{\hbox{49}}5\underbrace{\hbox{0...0}}_{\hbox{99}}1$ is not the cube of any integer.

Prove that there exists a unique point $M$ on the side $AD$ of a convex quadrilateral $ABCD$ such that \[\sqrt{S_{ABM}}+\sqrt{S_{CDM}} = \sqrt{S_{ABCD}}\] if and only if $AB\parallel CD$.

Side $\overline{AB}$ of $\triangle ABC$ has length $10$. The bisector of angle $A$ meets $\overline{BC}$ at $D$, and $CD = 3$. The set of all possible values of $AC$ is an open interval $(m,n)$. What is $m+n$? $ \textbf{(A) }16 \qquad \textbf{(B) }17 \qquad \textbf{(C) }18 \qquad \textbf{(D) }19 \qquad \textbf{(E) }20 \qquad $

Let $AB$ be a diameter of the circle $\Omega$ with center $O{}$. The points $C, D, X$ and $Y{}$ are chosen on $\Omega$ so that the segments $CX$ and $DX$ intersect the segment $AB$ at points symmetric with respect to $O{}$, and $XY\parallel AB$. Let the lines $AB{}$ and $CD{}$ intersect at the point $E$. Prove that the tangent to $\Omega$ through $Y{}$ passes through $E{}$.

$ABC$ is an isosceles triangle, with $AB=AC$. $D$ is a moving point such that $AD\parallel BC$, $BD>CD$. Moving point $E$ is on the arc of $BC$ in circumcircle of $ABC$ not containing $A$, such that $EB<EC$. Ray $BC$ contains point $F$ with $\angle ADE=\angle DFE$. If ray $FD$ intersects ray $BA$ at $X$, and intersects ray $CA$ at $Y$, prove that $\angle XEY$ is a fixed angle.

Two circles $\Omega_{1}$ and $\Omega_{2}$ touch internally the circle $\Omega$ in M and N and the center of $\Omega_{2}$ is on $\Omega_{1}$. The common chord of the circles $\Omega_{1}$ and $\Omega_{2}$ intersects $\Omega$ in $A$ and $B$. $MA$ and $MB$ intersects $\Omega_{1}$ in $C$ and $D$. Prove that $\Omega_{2}$ is tangent to $CD$.

Three spheres each of unit radius have centers $P,Q,R$ with the property that the center of each sphere lies on the surface of the other two spheres. Let $C$ denote the cylinder with cross-section $PQR$ (the triangular lamina with vertices $P,Q,R$) and axis perpendicular to $PQR$. Let $M$ denote the space which is common to the three spheres and the cylinder $C$, and suppose the mass density of $M$ at a given point is the distance of the point from $PQR$. Determine the mass of $M$.

Let $ABC$ be a triangle with circumcentre $O$ and centroid $G$. Let $M$ be the midpoint of $BC$ and $N$ the reflection of $M$ across $O$. Prove that $NO = NA$ if and only if $\angle AOG = 90^{\circ}$. [i]Proposed by Pranjal Srivastava[/i]

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.

Construct a triangle, given the feet of its altitudes. Express the sides of a triangle $Y$ in terms of the sides of the triangle $X$ formed by the feet of the altitudes of $Y$.

Triangle $ABC$ is said to be perpendicular to triangle $DEF$ if the perpendiculars from $A$ to $EF$,from $B$ to $FD$ and from $C$ to $DE$ are concurrent.Prove that if $ABC$ is perpendicular to $DEF$,then $DEF$ is perpendicular to $ABC$