$ABC$ is a triangle with sides $1$, $2$ and $\sqrt3$. Determine the smallest possible area of an equilateral triangle with a vertex on each side of triangle $ABC$.

[b]U[/b]ndecillion years ago in a galaxy far, far away, there were four space stations in the three-dimensional space, each pair spaced 1 light year away from each other. Admiral Ackbar wanted to establish a base somewhere in space such that the sum of squares of the distances from the base to each of the stations does not exceed 15 square light years. (The sizes of the space stations and the base are negligible.) Determine the volume, in cubic light years, of the set of all possible locations for the Admiral’s base.

[b]rolling cube[/b] $a$,$b$ and $c$ are natural numbers. we have a $(2a+1)\times (2b+1)\times (2c+1)$ cube. this cube is on an infinite plane with unit squares. you call roll the cube to every side you want. faces of the cube are divided to unit squares and the square in the middle of each face is coloured (it means that if this square goes on a square of the plane, then that square will be coloured.) prove that if any two of lengths of sides of the cube are relatively prime, then we can colour every square in plane. time allowed for this question was 1 hour.

Let the area and the perimeter of a cyclic quadrilateral $C$ be $A_C$ and $P_C$, respectively. If the area and the perimeter of the quadrilateral which is tangent to the circumcircle of $C$ at the vertices of $C$ are $A_T$ and $P_T$ , respectively, prove that $\frac{A_C}{A_T} \geq \left (\frac{P_C}{P_T}\right )^2$.

$ABCD$ is a cyclic quadrilateral inscribed in the circle $\omega$. Let $AB \cap CD = E$, $AD \cap BC = F$. Let $\omega_1, \omega_2$ be the circumcircles of $AEF, CEF$, respectively. Let $\omega \cap \omega_1 = G$, $\omega \cap \omega_2 = H$. Show that $AC, BD, GH$ are concurrent. [i]Proposed by Yang Liu[/i]

In the acute triangle $ABC$, the bisectors of $A,B$ and $C$ intersect the circumcircle again at $A_1,B_1$ and $C_1$, respectively. Let $M$ be the point of intersection of $AB$ and $B_1C_1$, and let $N$ be the point of intersection of $BC$ and $A_1B_1$. Prove that $MN$ passes through the incentre of $\triangle ABC$.

Let $ABC$ be a triangle such that $$S^2=2R^2+8Rr+3r^2$$ Then prove that $\frac Rr=2$ or $\frac Rr\ge\sqrt2+1$. [i]Proposed by Marian Cucoanoeş and Marius Drăgan[/i]

Suppose $ABC$ is a triangle with $M$ the midpoint of $BC$. Suppose that $AM$ intersects the incircle at $K,L$. We draw parallel line from $K$ and $L$ to $BC$ and name their second intersection point with incircle $X$ and $Y$. Suppose that $AX$ and $AY$ intersect $BC$ at $P$ and $Q$. Prove that $BP=CQ$.

There are $n$ points in a plane. Prove that there exist a point $O$ (not necessarily from the given $n$) such that on each side of an arbitrary line, through $O$, lie at least $\frac{n}{3}$ points (including the points on the line).

Positive integers are written on all the faces of a cube, one on each. At each corner of the cube, the product of the numbers on the faces that meet at the vertex is written. The sum of the numbers written on the corners is 2004. If T denotes the sum of the numbers on all the faces, find the possible values of T.

In the figure, the area of square WXYZ is $25 \text{cm}^2$. The four smaller squares have sides 1 cm long, either parallel to or coinciding with the sides of the large square. In $\Delta ABC$, $AB = AC$, and when $\Delta ABC$ is folded over side BC, point A coincides with O, the center of square WXYZ. What is the area of $\Delta ABC$, in square centimeters? [asy] defaultpen(fontsize(8)); size(225); pair Z=origin, W=(0,10), X=(10,10), Y=(10,0), O=(5,5), B=(-4,8), C=(-4,2), A=(-13,5); draw((-4,0)--Y--X--(-4,10)--cycle); draw((0,-2)--(0,12)--(-2,12)--(-2,8)--B--A--C--(-2,2)--(-2,-2)--cycle); dot(O); label("$A$", A, NW); label("$O$", O, NE); label("$B$", B, SW); label("$C$", C, NW); label("$W$",W , NE); label("$X$", X, N); label("$Y$", Y, N); label("$Z$", Z, SE); [/asy] $ \textbf{(A)}\ \frac{15}4\qquad\textbf{(B)}\ \frac{21}4\qquad\textbf{(C)}\ \frac{27}4\qquad\textbf{(D)}\ \frac{21}2\qquad\textbf{(E)}\ \frac{27}2$

An angle bisector $AD$ was drawn in triangle $ABC$. It turned out that the center of the inscribed circle of triangle $ABC$ coincides with the center of the inscribed circle of triangle $ABD$. Find the angles of the original triangle.

Frist Campus Center is located $1$ mile north and $1$ mile west of Fine Hall. The area within $5$ miles of Fine Hall that is located north and east of Frist can be expressed in the form $\frac{a}{b} \pi - c$, where $a, b, c$ are positive integers and $a$ and $b$ are relatively prime. Find $a + b + c$.

Let $ M $ be a point in the interior of a triangle $ ABC, $ let $ D $ be the intersection of $ AM $ with $ BC, $ let $ E $ be the intersection of $ M $ with AC, let $ F $ be the intersection of $ CM $ with $ AB. $ Knowing that the expression $$ \frac{MA}{MD}\cdot \frac{MB}{ME}\cdot \frac{MC}{MF} $$ is minimized, describe the point $ M. $

An acute triangle has unit area. Show that there is a point inside the triangle whose distance from each of the vertices is at least $\frac{2}{\sqrt[4]{27}}$.

In $3$-dimensional space there is given a point $O$ and a finite set $A$ of segments with the sum of lengths equal to $1988$. Prove that there exists a plane disjoint from $A$ such that the distance from it to $O$ does not exceed $574$.

The nine horizontal and nine vertical lines on an $8\times8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Define $A = (1, 0, 0)$, $B = (0, 1, 0)$, and $P$ as the set of all points $(x, y, z)$ such that $x+y+z = 0$. Let $P$ be the point on $P$ such that $d = AP + P B$ is minimized. Find $d^2$.

Circles $K_1$ and $K_2$ are externally tangent to each other at $A$ and are internally tangent to a circle $K$ at $A_1$ and $A_2$ respectively. The common tangent to $K_1$ and $K_2$ at $A$ meets $K$ at point $P$. Line $PA_1$ meets $K_1$ again at $B_1$ and $PA_2$ meets $K_2$ again at $B_2$. Show that $B_1B_2$ is a common tangent of $K_1$ and $K_2$.

Find the circumradius of the triangle with side lengths $104$, $112$, and $120$.

A circle $\omega$ not passing through any vertex of $\triangle ABC$ intersects each of the segments $AB$, $BC$, $CA$ in 2 distinct points. Prove that the incenter of $\triangle ABC$ lies inside $\omega$. [i]Evan O' Dorney.[/i]

For a given cyclic quadrilateral $ABCD$, let $I$ be a variable point on the diagonal $AC$ such that $I$ and $A$ are on the same side of the diagonal $BD$. Assume $E,F$ lie on the diagonal $BD$ such that $IE\parallel AB$ and $IF\parallel AD$. Show that $\angle BIE =\angle DCF $

Show that the cube roots of three distinct prime numbers cannot be three terms (not necessarily consecutive) of an arithmetic progression.

Points $A$, $B$, and $C$ lie on a semicircle with diameter $\overline{PQ}$ such that $AB = 3$, $AC = 4$, $BC = 5$, and $A$ is on $\overline{PQ}$. Given $\angle PAB = \angle QAC$, compute the area of the semicircle.

For a unit circle $O$, arrange points $A,B,C,D$ and $E$ in that order evenly along $O$'s circumference. For each of those points, draw the arc centered at that point inside O from the point to its left to the point to its right. Denote the outermost intersections of these arcs as $A', B', C', D'$ and $E'$, where the prime of any point is opposite the point. The length of $AC'$ can be written as an expression $f(x)$, where $f$ is a trigonometric function. Find this expression.