Consider the shape shown below, formed by gluing together the sides of seven congruent regular hexagons. The area of this shape is partitioned into $21$ quadrilaterals, all of whose side lengths are equal to the side length of the hexagon and each of which contains a $60^{\circ}$ angle. In how many ways can this partitioning be done? (The quadrilaterals may contain an internal boundary of the seven hexagons.) [asy] draw(origin--origin+dir(0)--origin+dir(0)+dir(60)--origin+dir(0)+dir(60)+dir(0)--origin+dir(0)+dir(60)+dir(0)+dir(60)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)+dir(180)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)+dir(180)+dir(240)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)+dir(180)+dir(240)+dir(300)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)+dir(180)+dir(240)+dir(300)+dir(240)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)+dir(180)+dir(240)+dir(300)+dir(240)+dir(300)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)+dir(180)+dir(240)+dir(300)+dir(240)+dir(300)+dir(0)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)+dir(180)+dir(240)+dir(300)+dir(240)+dir(300)+dir(0)+dir(300)--cycle); draw(2*dir(60)+dir(120)+dir(0)--2*dir(60)+dir(120)+2*dir(0),dashed); draw(2*dir(60)+dir(120)+dir(60)--2*dir(60)+dir(120)+2*dir(60),dashed); draw(2*dir(60)+dir(120)+dir(120)--2*dir(60)+dir(120)+2*dir(120),dashed); draw(2*dir(60)+dir(120)+dir(180)--2*dir(60)+dir(120)+2*dir(180),dashed); draw(2*dir(60)+dir(120)+dir(240)--2*dir(60)+dir(120)+2*dir(240),dashed); draw(2*dir(60)+dir(120)+dir(300)--2*dir(60)+dir(120)+2*dir(300),dashed); draw(dir(60)+dir(120)--dir(60)+dir(120)+dir(0)--dir(60)+dir(120)+dir(0)+dir(60)--dir(60)+dir(120)+dir(0)+dir(60)+dir(120)--dir(60)+dir(120)+dir(0)+dir(60)+dir(120)+dir(180)--dir(60)+dir(120)+dir(0)+dir(60)+dir(120)+dir(180)+dir(240)--dir(60)+dir(120)+dir(0)+dir(60)+dir(120)+dir(180)+dir(240)+dir(300),dashed); [/asy]

Let $ABCD$ be a cyclic quadrilateral. Points $E$ and $F$ lie on the sides $AD$ and $CD$ in such a way that $AE = BC$ and $AB = CF$. Let $M$ be the midpoint of $EF$. Prove that $\angle AMC = 90^{\circ}$.

Find all perfect powers whose last $ 4$ digits are $ 2,0,0,8$, in that order.

Exists a triangle whose three altitudes have lengths $1, 2$ and $3$ respectively?

Let $ABCD$ be a convex quadrilateral such that the circle with diameter $AB$ is tangent to the line $CD$, and the circle with diameter $CD$ is tangent to the line $AB$. Prove that the two intersection points of these circles and the point $AC \cap BD$ are collinear.

A set of points is marked on the plane, with the property that any three marked points can be covered with a disk of radius $1$. Prove that the set of all marked points can be covered with a disk of radius $1$.

Given convex quadrilateral $ABCD$. We construct the similar triangles $APB, BQC, CRD, DSA$ outside $ABCD$ so that $\angle PAB = \angle QBC = \angle RCD = \angle SDA, \angle PBA = \angle QCB = \angle RDC = \angle SAD$. Prove that if $PQRS$ is a parallelogram, so is $ABCD$.

In obtuse triangle $ABC$, with the obtuse angle at $A$, let $D$, $E$, $F$ be the feet of the altitudes through $A$, $B$, $C$ respectively. $DE$ is parallel to $CF$, and $DF$ is parallel to the angle bisector of $\angle BAC$. Find the angles of the triangle.

Let $A$ be a point on a circle with center $O$ and $B$ be the midpoint of $[OA]$. Let $C$ and $D$ be points on the circle such that they lie on the same side of the line $OA$ and $\widehat{CBO} = \widehat{DBA}$. Show that the reflection of the midpoint of $[CD]$ over $B$ lies on the circle.

In a triangle $ABC$ the midpoints of $BC, CA$ and $AB$ are $D, E$ and $F$, respectively. Prove that the circumcircles of triangles $AEF, BFD$ and $CDE$ intersect all in one point.

In an acute-angled triangle $ABC$, $AM$ is a median, $AL$ is a bisector and $AH$ is an altitude ($H$ lies between $L$ and $B$). It is known that $ML=LH=HB$. Find the ratios of the sidelengths of $ABC$.

In the isosceles triangle $ABC$ ($AB=AC$), let $l$ be a line parallel to $BC$ through $A$. Let $D$ be an arbitrary point on $l$. Let $E,F$ be the feet of perpendiculars through $A$ to $BD,CD$ respectively. Suppose that $P,Q$ are the images of $E,F$ on $l$. Prove that $AP+AQ\le AB$ [i]Proposed by Morteza Saghafian[/i]

Let $ABCD$ be a quadrilateral inscribed in circle $\omega$. Define $E = AA \cap CD$, $F = AA \cap BC$, $G = BE \cap \omega$, $H = BE \cap AD$, $I = DF \cap \omega$, and $J = DF \cap AB$. Prove that $GI$, $HJ$, and the $B$-symmedian are concurrent. [i]Proposed by Robin Park[/i]

Let a non-equilateral triangle $ ABC$ and $ AD,BE,CF$ are its altitudes. On the rays $ AD,BE,CF,$ respectively, let $ A',B',C'$ such that $ \frac {AA'}{AD} \equal{} \frac {BB'}{BE} \equal{} \frac {CC'}{CF} \equal{} k$. Find all values of $ k$ such that $ \triangle A'B'C'\sim\triangle ABC$ for any non-triangle $ ABC.$

Determine the area of the region defined by [i]x[/i]²+[i]y[/i]²≤[i]π[/i]² and [i]y[/i] ≥ sin [i]x[/i].

In a triangle $ABC$ with an angle $\angle CAB =30^o$ draw median $CD$. If the formed $\vartriangle ACD$ is isosceles, find tan $\angle DCB$.

For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$? [i]Proposed by Andre Arslan[/i]

The parabola $y = x^2$ intersects a circle at exactly two points $A$ and $B$. If their tangents at $A$ coincide, must their tangents at $B$ also coincide?

Let $ ABCD$ be an isosceles trapezium with $ AB \parallel{} CD$ and $ \bar{BC} \equal{} \bar{AD}.$ The parallel to $ AD$ through $ B$ meets the perpendicular to $ AD$ through $ D$ in point $ X.$ The line through $ A$ drawn which is parallel to $ BD$ meets the perpendicular to $ BD$ through $ D$ in point $ Y.$ Prove that points $ C,X,D$ and $ Y$ lie on a common circle.

A small square $PQRS$ is contained in a big square. Produce $PQ$ to $A$, $QR$ to $B$, $RS$ to $C$ and $SP$ to $D$ so that $A$, $B$, $C$ and $D$ lie on the four sides of the large square in order, produced if necessary. Prove that $AC = BD$ and $AC \perp BD$.

A regular octagon $P$ is given whose incircle $k$ has diameter $1$. About $k$ is circumscribed a regular $16$-gon, which is also inscribed in $P$, cutting from $P$ eight isosceles triangles. To the octagon $P$, three of these triangles are added so that exactly two of them are adjacent and no two of them are opposite to each other. Every $11$-gon so obtained is said to be $P'$. Prove the following statement: Given a finite set $M$ of points lying in $P$ such that every two points of this set have a distance not exceeding $1$, one of the $11$-gons $P'$ contains all of $M$.

The sum of the areas of all triangles whose vertices are also vertices of a $1\times 1 \times 1$ cube is $m+\sqrt{n}+\sqrt{p}$, where $m$, $n$, and $p$ are integers. Find $m+n+p$.

Let $a,b,c \in \left[ \frac 12, 1 \right]$. Prove that \[ 2 \leq \frac{ a+b}{1+c} + \frac{ b+c}{1+a} + \frac{ c+a}{1+b} \leq 3 . \] [i]selected by Mircea Lascu[/i]

Let $O$ be the point $(0,0)$. Let $A$, $B$, $C$ be three points in the plane such that $AO=15$, $BO = 15$, and $CO = 7$, and such that the area of triangle $ABC$ is maximal. What is the length of the shortest side of $ABC$?

Four different points $A,B,C$ and $D$ lie in a plane. No three of these points lie on a single straight line. Describe the construction of a square $PQRS$ such that on each of the sides of $PQRS$, or the extensions , lies one of the points $A, B, C$ and $D$.