Let $ ABCD$ be a rhombus with $ \angle BAD \equal{} 60^{\circ}$. Points $ S$ and $ R$ are chosen inside the triangles $ ABD$ and $ DBC$, respectively, such that $ \angle SBR \equal{} \angle RDS \equal{} 60^{\circ}$. Prove that $ SR^2\geq AS\cdot CR$.

For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. How many different values of $p<2015$ are possible? $\textbf{(A) }30\qquad\textbf{(B) }31\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

Find the area of rhombus $ABCD$ given that the radii of the circles circumscribed around triangles $ABD$ and $ACD$ are $12.5$ and $25$, respectively.

Let $ABCD$ be a cyclic quadrilateral.$E$ is the midpoint of $(AC)$ and $F$ is the midpoint of $(BD)$ {$G$}=$AB\cap CD$ and {$H$}=$AD\cap BC$. a)Prove that the intersections of the angle bisector of $\angle{AHB}$ and the sides $AB$ and $CD$ and the intersections of the angle bisector of$\angle{AGD}$ with $BC$ and $AD$ are the verticles of a rhombus b)Prove that the center of this rhombus lies on $EF$

A square sheet of paper $ABCD$ is so folded that $B$ falls on the mid point of $M$ of $CD$. Prove that the crease will divide $BC$ in the ration $5 : 3$.

Let $ABC$ be an acute-angled triangle, with $AC \neq BC$ and let $O$ be its circumcenter. Let $P$ and $Q$ be points such that $BOAP$ and $COPQ$ are parallelograms. Show that $Q$ is the orthocenter of $ABC$.

Find the least positive integer $n$ ($n\geq 3$), such that among any $n$ points (no three are collinear) in the plane, there exist three points which are the vertices of a non-isoscele triangle.

A rhombus is inscribed in triangle $ ABC$ in such a way that one of its vertices is $ A$ and two of its sides lie along $ AB$ and $ AC$. If $ \overline{AC} \equal{} 6$ inches, $ \overline{AB} \equal{} 12$ inches, and $ \overline{BC} \equal{} 8$ inches, the side of the rhombus, in inches, is: $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 3 \frac {1}{2} \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

A tangent $l$ to the circle inscribed in a rhombus meets its sides $AB$ and $BC$ at points $E$ and $F$ respectively. Prove that the product $AE\cdot CF$ is independent of the choice of $l$.

Let $ABCD$ be a cyclic quadrilateral. Points $K, L, M, N$ are chosen on $AB, BC, CD, DA$ such that $KLMN$ is a rhombus with $KL \parallel AC$ and $LM \parallel BD$. Let $\omega_A, \omega_B, \omega_C, \omega_D$ be the incircles of $\triangle ANK, \triangle BKL, \triangle CLM, \triangle DMN$. Prove that the common internal tangents to $\omega_A$, and $\omega_C$ and the common internal tangents to $\omega_B$ and $\omega_D$ are concurrent.

Let a hexagone with a diameter $D$ be given and let $d>\frac D 2.$ On each side of the hexagon one constructs a isosceles triangle with two equal sides of length $d$. Prove that the sum of the areas of those isoscele triangles is greater than the area of a rhombus with side lengths $d$ and a diagonal of length $D$. (The diameter of a polygon is the maximum of the lengths of all its sides and diagonals.)

Given four points $ A_1, A_2, A_3, A_4$ in the plane, no three collinear, such that \[ A_1A_2 \cdot A_3 A_4 \equal{} A_1 A_3 \cdot A_2 A_4 \equal{} A_1 A_4 \cdot A_2 A_3, \] denote by $ O_i$ the circumcenter of $ \triangle A_j A_k A_l$ with $ \{i,j,k,l\} \equal{} \{1,2,3,4\}.$ Assuming $ \forall i A_i \neq O_i ,$ prove that the four lines $ A_iO_i$ are concurrent or parallel. [i]Nikolai Ivanov Beluhov, Bulgaria[/i]

It is allowed to perform the following transformations in the plane with any integers $a$: (1) Transform every point $(x, y)$ to the corresponding point $(x + ay, y)$, (2) Transform every point $(x, y)$ to the corresponding point $(x, y + ax)$. Does there exist a non-square rhombus whose all vertices have integer coordinates and which can be transformed to: a) Vertices of a square, b) Vertices of a rectangle with unequal side lengths?

The two diagonals of a rhombus have lengths with ratio $3 : 4$ and sum $56$. What is the perimeter of the rhombus?

A regular tetrahedron is sectioned with a plane after a rhombus. Prove that the rhombus is square.

Two circles $ O, O'$ having same radius meet at two points, $ A,B (A \not = B) $. Point $ P,Q $ are each on circle $ O $ and $ O' $ $(P \not = A,B ~ Q\not = A,B )$. Select the point $ R $ such that $ PAQR $ is a parallelogram. Assume that $ B, R, P, Q $ is cyclic. Now prove that $ PQ = OO' $.

2. In a square $ABCD$, let $P$ be a point in the side $CD$, different from $C$ and $D$. In the triangle $ABP$, the altitudes $AQ$ and $BR$ are drawn, and let $S$ be the intersection point of lines $CQ$ and $DR$. Show that $\angle ASB=90$.

In the triangle $ABC$ the angle bisectors $AL$ and $BT$ are drawn, which intersect at the point $I$, and their extensions intersect the circle circumscribed around the triangle $ABC$ at the points $E$ and $D$ respectively. The segment $DE$ intersects the sides $AC$ and $BC$ at the points $F$ and $K$, respectively. Prove that: a) quadrilateral $IKCF$ is rhombus; b) the side of this rhombus is $\sqrt {DF \cdot EK}$. (Rozhkova Maria)

The quadrilateral $ABCD$ is a rhombus with acute angle at $A.$ Points $M$ and $N$ are on segments $\overline{AC}$ and $\overline{BC}$ such that $|DM| = |MN|.$ Let $P$ be the intersection of $AC$ and $DN$ and let $R$ be the intersection of $AB$ and $DM.$ Prove that $|RP| = |PD|.$

Let $ABCD$ be a convex quadrilateral such that $AC \perp BD$. (a) Prove that $AB^2 + CD^2 = BC^2 + DA^2$. (b) Let $PQRS$ be a convex quadrilateral such that $PQ = AB$, $QR = BC$, $RS = CD$ and $SP = DA$. Prove that $PR \perp QS$.

In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on different sides of $AEDB$). Let $\varphi$ be the non-obtuse angle of the rhombus. Prove that $\varphi \le \max \{ \angle BAC, \angle ABC \}$.

Let $\Gamma$ be a circle with center $O$. Consider the points $A$, $B$, $C$, $D$, $E$ and $F$ in $\Gamma$, with $D$ and $E$ in the (minor) arc $BC$ and $C$ in the (minor) arc $EF$, such that $DEFO$ is a rhombus and $\vartriangle ABC$ It is equilateral. Show that $\overleftrightarrow{BD}$ and $\overleftrightarrow{CE}$ are perpendicular.

The diagonals of a convex quadrilateral divide it into four triangles. The radii of the circles circumscribed around these triangles are equal. Can such a property have a quadrilateral other than: a) parallelogram, b) rhombus? (Sharygin Igor)

A circle cuts each side of a rhombus twice thus dividing each side into three segments. Let us go around the perimeter of the rhombus clockwise beginning at a vertex and paint these segments successively in red, white and blue. Prove that the sum of lengths of the blue segments equals that of the red ones. (V Proizvolov)

Four circles of radius $ 1$ are each tangent to two sides of a square and externally tangent to a circle of radius $ 2$, as shown. What is the area of the square? [asy]unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); real h=3*sqrt(2)/2; pair O0=(0,0), O1=(h,h), O2=(-h,h), O3=(-h,-h), O4=(h,-h); pair X=O0+2*dir(30), Y=O2+dir(45); draw((-h-1,-h-1)--(-h-1,h+1)--(h+1,h+1)--(h+1,-h-1)--cycle); draw(Circle(O0,2)); draw(Circle(O1,1)); draw(Circle(O2,1)); draw(Circle(O3,1)); draw(Circle(O4,1)); draw(O0--X); draw(O2--Y); label("$2$",midpoint(O0--X),NW); label("$1$",midpoint(O2--Y),SE);[/asy]$ \textbf{(A)}\ 32 \qquad \textbf{(B)}\ 22 \plus{} 12\sqrt {2}\qquad \textbf{(C)}\ 16 \plus{} 16\sqrt {3}\qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 36 \plus{} 16\sqrt {2}$