A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called [i]unstuck[/i] if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest perimeter has the form $\sqrt{N}$, for a positive integer $N$. Find $N$.

Four balls of radius $1$ are placed in space so that each of them touches the other three. What is the radius of the smallest sphere containing all of them?

Let $\mathcal S$ denote the set of points inside or on the border of a triangle $ABC$, without a fixed point $T$ inside the triangle. Show that $\mathcal S$ can be partitioned into disjoint closed segemnts. [i]Yugoslavia[/i]

Determine a necessary and sufficient condition for the affixes of three complex numbers $z_1$ , $z_2$ and $z_3$ are the vertices of an equilateral triangle.

Let $I$ be the incenter of a non-isosceles triangle $ABC$. The line $AI$ intersects the circumcircle of the triangle $ABC$ at $A$ and $D$. Let $M$ be the middle point of the arc $BAC$. The line through the point $I$ perpendicular to $AD$ intersects $BC$ at $F$. The line $MI$ intersects the circle $BIC$ at $N$. Prove that the line $FN$ is tangent to the circle $BIC$.

The extensions of two opposite sides of the convex quadrilateral intersect and form an angle of $20^o$ , the extensions of the other two sides also intersect and form an angle of $20^o$. It is known that exactly one angle of the quadrilateral is $80^o$. Find all of its other angles.

$ABCD$ is a rectangle. $E$ is a point on $AB$ between $A$ and $B$, and $F$ is a point on $AD$ between $A$ and $D$. The area of the triangle $EBC$ is $16$, the area of the triangle $EAF$ is $12$ and the area of the triangle $FDC$ is 30. Find the area of the triangle $EFC$.

A chord $ST$ of constant length slides around a semicircle with diameter $AB$. $M$ is the midpoint of $ST$ and $P$ is the foot of the perpendicular from $S$ to $AB$. Prove that $\angle SPM$ is constant for all positions of $ST$.

The $A,B,C$ and $D$ points (from left to right) belong to the straight line. Prove that every point $E$, that doesn't belong to the line satisfy: $$|AE| + |ED| + | |AB| - |CD| | > |BE| + |CE|$$

Let $ABC$ be a triangle. Suppose that $D$, $E$, and $F$ are points on segments $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ respectively such that triangles $AEF$, $BFD$, and $CDE$ have equal inradii. Prove that the sum of the inradii of $\triangle AEF$ and $\triangle DEF$ is equal to the inradius of $\triangle ABC$. [i]Aprameya Tripathy[/i]

A dart is thrown at a square dartboard of side length $2$ so that it hits completely randomly. What is the probability that it hits closer to the center than any corner, but within a distance $1$ of a corner?

Let $ ABCD$ be a convex quadrilateral such that $ AB$ and $ CD$ are not parallel and $ AB\equal{}CD$. The midpoints of the diagonals $ AC$ and $ BD$ are $ E$ and $ F$, respectively. The line $ EF$ meets segments $ AB$ and $ CD$ at $ G$ and $ H$, respectively. Show that $ \angle AGH \equal{} \angle DHG$.

Let $ d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $ n$ vertices (where $ n>3$). Let $ p$ be its perimeter. Prove that: \[ n\minus{}3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n\plus{}1\over 2}\Bigr]\minus{}2,\] where $ [x]$ denotes the greatest integer not exceeding $ x$.

Inside triangle $ABC$ there are three circles $k_1, k_2, k_3$ each of which is tangent to two sides of the triangle and to its incircle $k$. The radii of $k_1, k_2, k_3$ are $1, 4$, and $9$. Determine the radius of $k.$

[4] Let $ABCDEF$ be a regular hexagon. Let $P$ be the circle inscribed in $\triangle{BDF}$. Find the ratio of the area of circle $P$ to the area of rectangle $ABDE$.

$ABCD$ is a rectangle with $AB = CD = 2$. A circle centered at $O$ is tangent to $BC$, $CD$, and $AD$ (and hence has radius $1$). Another circle, centered at $P$, is tangent to circle $O$ at point $T$ and is also tangent to $AB$ and $BC$. If line $AT$ is tangent to both circles at $T$, find the radius of circle $P$.

Circle $k$ with center $O$ is touched from inside by two circles in points $S$ and $T,$ respectively. Let those two circles intersect at points $M$ and $N$, such that $N$ is closer to line $ST$. Prove that $OM$ and $MN$ are perpendicular iff $S$, $N$ and $T$ are collinear

16. Create a cube $C_1$ with edge length $1$. Take the centers of the faces and connect them to form an octahedron $O_1$. Take the centers of the octahedron’s faces and connect them to form a new cube $C_2$. Continue this process infinitely. Find the sum of all the surface areas of the cubes and octahedrons. 17. Let $p(x) = x^2 - x + 1$. Let $\alpha$ be a root of $p(p(p(p(x)))$. Find the value of $$(p(\alpha) - 1)p(\alpha)p(p(\alpha))p(p(p(\alpha))$$ 18. An $8$ by $8$ grid of numbers obeys the following pattern: 1) The first row and first column consist of all $1$s. 2) The entry in the $i$th row and $j$th column equals the sum of the numbers in the $(i - 1)$ by $(j - 1)$ sub-grid with row less than i and column less than $j$. What is the number in the 8th row and 8th column?

Let $ ABC$ a triangle. Let $D$ on $AB$ and $E$ on $AC$ such that $DE||BC$. Let line $DE$ intersect circumcircle of $ABC$ at two distinct points $F$ and $G$ so that line segments $BF$ and $CG$ intersect at P. Let circumcircle of $GDP$ and $FEP$ intersect again at $Q$. Prove that $A, P, Q$ are collinear.

Let $ABC$ be an acute-angled triangle, $C'$ and $A'$ be arbitrary points on the sides $AB$ and $BC$ respectively, and $B'$ be the midpoint of the side $AC$. (a) Prove that the area of triangle $A'B'C'$ is at most half the area of triangle $ABC$. (b) Prove that the area of triangle $A'B'C'$ is equal to one fourth of the area of triangle $ABC$ if and only if at least one of the points $A'$, $C'$ is the midpoint of the corresponding side. (E Cherepanov)

On the plane are chosen $n \ge 3$ points, not all on the same line. Drawing all lines passing through two of these points one obtains k different lines. Prove that $k \ge n$.

Can a regular triangle be placed inside a regular hexagon in such a way that all vertices of the triangle were seen from each vertex of the hexagon? (Point $A$ is seen from $B$, if the segment $AB$ dots not contain internal points of the triangle.)

In a scalene triangle $ ABC$ the altitudes $ AA_{1}$ and $ CC_{1}$ intersect at $ H, O$ is the circumcenter, and $ B_{0}$ the midpoint of side $ AC$. The line $ BO$ intersects side $ AC$ at $ P$, while the lines $ BH$ and $ A_{1}C_{1}$ meet at $ Q$. Prove that the lines $ HB_{0}$ and $ PQ$ are parallel.

Given a triangle $ \triangle{ABC} $ and a point $ O $. $ X $ is a point on the ray $ \overrightarrow{AC} $. Let $ X' $ be a point on the ray $ \overrightarrow{BA} $ so that $ \overline{AX} = \overline{AX_{1}} $ and $ A $ lies in the segment $ \overline{BX_{1}} $. Then, on the ray $ \overrightarrow{BC} $, choose $ X_{2} $ with $ \overline{X_{1}X_{2}} \parallel \overline{OC} $. Prove that when $ X $ moves on the ray $ \overrightarrow{AC} $, the locus of circumcenter of $ \triangle{BX_{1}X_{2}} $ is a part of a line.

In acute triangle $ABC, \angle A = 45^o$. Points $O,H$ are the circumcenter and the orthocenter of $ABC$, respectively. $D$ is the foot of altitude from $B$. Point $X$ is the midpoint of arc $AH$ of the circumcircle of triangle $ADH$ that contains $D$. Prove that $DX = DO$. Proposed by Fatemeh Sajadi