There is a rhombus with acute angle $b$ and side $a$. Two parallel lines, the distance between which is equal to the height of the rhombus, intersect all four sides of the rhombus. What can be the sum of the perimeters of two triangles cut off from a rhombus by straight lines? (These two triangles lie outside the strip between parallel lines.)

Let $P$ be a regular $2n$-sided polygon. A [b]rhombus-ulation[/b] of $P$ is dividing $P$ into rhombuses such that no two intersect and no vertex of any rhombus is on the edge of other rhombuses or $P$. (a) Prove that number of rhombuses is a function of $n$. Find the value of this function. Also find the number of vertices and edges of the rhombuses as a function of $n$. (b) Prove or disprove that there always exists an edge $e$ of $P$ such that by erasing all the segments parallel to $e$ the remaining rhombuses are connected. (c) Is it true that each two rhombus-ulations can turn into each other using the following algorithm multiple times? Algorithm: Take a hexagon -not necessarily regular- consisting of 3 rhombuses and re-rhombus-ulate the hexagon. (d) Let $f(n)$ be the number of ways to rhombus-ulate $P$. Prove that:\[\Pi_{k=1}^{n-1} ( \binom{k}{2} +1) \leq f(n) \leq \Pi_{k=1}^{n-1} k^{n-k} \]

A line $\ell$ passes through vertex $C$ of the rhombus $ABCD$ and meets the extensions of $AB, AD$ at points $X,Y$. Lines $DX, BY$ meet $(AXY)$ for the second time at $P,Q$. Prove that the circumcircle of $\triangle PCQ$ is tangent to $\ell$ [i]A. Kuznetsov[/i]

Two different points $A, M$ are given in a plane, $AM = d > 0$. Let a number $v > 0$ be given. Construct a rhombus $ABCD$ with the height of length $v$ and $M$ being a midpoint of $BC$. Discuss conditions of solvability and determine number of solutions. Can the resulting quadrilateral $ABCD$ be a square?

In the following, a rhombus is one with edge length $1$ and interior angles $60^o$ and $120^o$ . Now let $n$ be a natural number and $H$ a regular hexagon with edge length $n$, which is covered with rhombuses without overlapping has been. The rhombuses then appear in three different orientations. Prove that whatever the overlap looks exactly, each of these three orientations can be viewed at the same time.

Points $ K,L,M,N$ are repectively the midpoints of sides $ AB,BC,CD,DA$ in a convex quadrliateral $ ABCD$.Line $ KM$ meets dioganals $ AC$ and $ BD$ at points $ P$ and $ Q$,respectively.Line $ LN$ meets dioganals $ AC$ and $ BD$ at points $ R$ and $ S$,respectively. Prove that if $ AP\cdot PC\equal{}BQ\cdot QD$,then $ AR\cdot RC\equal{}BS\cdot SD$.

It is known that $\triangle ABC$ is an acute triangle. Let $C'$ be the foott of the perpendicular from $C$ to $AB$, and $D$, $E$ two distinct points on $CC'$. The feet of the perpendiculars from $D$ to $AC$ and $BC$ are $F$ and $G$, respectively. Show that if $DGEF$ is a parallelogram then $ABC$ is isosceles.

$(YUG 3)$ Let four points $A_i (i = 1, 2, 3, 4)$ in the plane determine four triangles. In each of these triangles we choose the smallest angle. The sum of these angles is denoted by $S.$ What is the exact placement of the points $A_i$ if $S = 180^{\circ}$?

A rhombus $ABCD$ is given with $\angle BAD = 60^o$ . Point $P$ lies inside the rhombus such that $BP = 1$, $DP = 2$, $CP = 3$. Determine the length of the segment $AP$.

Each of two different lines parallel to the the axis $Ox$ have exactly two common points on the graph of the function $f(x)=x^3+ax^2+bx+c$. Let $\ell_1$ and $\ell_2$ be two lines parallel to $Ox$ axis which meet the graph of $f$ in points $K_1, K_2$ and $K_3, K_4$, respectively. Prove that the quadrilateral formed by $K_1, K_2, K_3$ and $ K_4$ is a rhombus if and only if its area is equal to $6$ units.

Regular hexagon is divided to equal rhombuses, with sides, parallels to hexagon sides. On the three sides of the hexagon, among which there are no neighbors, is set directions in order of traversing the hexagon against hour hand. Then, on each side of the rhombus, an arrow directed just as the side of the hexagon parallel to this side. Prove that there is not a closed path going along the arrows.

Let $ABCD$ be a rhombus of sides $AB = BC = CD= DA = 13$. On the side $AB$ construct the rhombus $BAFE$ outside $ABCD$ and such that the side $AF$ is parallel to the diagonal $BD$ of $ABCD$. If the area of $BAFE$ is equal to $65$, calculate the area of $ABCD$.

Two players, $A$ and $B$, play a game on a board which is a rhombus of side $n$ and angles of $60^{\circ}$ and $120^{\circ}$, divided into $2n^2$ equilateral triangles, as shown in the diagram for $n=4$. $A$ uses a red token and $B$ uses a blue token, which are initially placed in cells containing opposite corners of the board (the $60^{\circ}$ ones). In turns, players move their token to a neighboring cell (sharing a side with the previous one). To win the game, a player must either place his token on the cell containing the other player's token, or get to the opposite corner to the one where he started. If $A$ starts the game, determine which player has a winning strategy.

We are given an equilateral triangle ABC with the length of its side equal to $1$. There are $n-1$ points on each side of the triangle $ABC$ that equally divide the side into $n$ segments. We draw all possible lines that pass through any two of all those $3(n-1)$ points such that they are parallel to one of three sides of triangle $ABC$. All such lines divide triangle $ABC$ into some lesser triangles whose vertices are called [i]nodes[/i]. We assign a real number for each [i]node[/i] such that the following conditions are satisfied: (I) real numbers $a,b,c$ are assigned to $A,B,C$ respectively; (II) for any rhombus that is consisted of two lesser triangles that share a common side, the sum of the numbers of vertices on its one diagonal is equal to that of vertices on the other diagonal. 1) Find the minimum distance between the [i]node[/i] with the maximal number to the [i]node[/i] with the minimal number; 2) Denote by $S$ the sum of the numbers of all [i]nodes[/i], find $S$.

Points $C_0$ and $B_0$ are the respective midpoints of sides $AB$ and $AC$ of a non-isosceles acute triangle $ABC$, $O$ is its circumscenter and $H$ is the orthocenter. Lines $BH$ and $OC_0$ intersect at $P$, while lines $CH$ and $OB_0$ intersect at $Q$. $OPHQ$ is rhombus. Prove that points $A$, $P$ and $Q$ are collinear. (Author: L. Emelyanov)

A convex quadrangle $ ABCD$ is inscribed in a circle with center $ O$. The angles $ AOB, BOC, COD$ and $ DOA$, taken in some order, are of the same size as the angles of the quadrangle $ ABCD$. Prove that $ ABCD$ is a square

Let $ABCD$ be a parallelogram and $M,N$ be points in the plane such that $C \in (AM)$ and $D \in (BN)$. Lines $NA,NC$ meet lines $MB,MD$ at points $E,F,G,H$. Show that points $E,F,G,H$ lie on a circle if and only if $ABCD$ is a rhombus.

Let $ABCDEF$ be a convex hexagon satisfying $AC=DF$, $CE=FB$ and $EA=BD$. Prove that the lines connecting the midpoints of opposite sides of the hexagon $ABCDEF$ intersect in one point.

If the diagonals of a quadrilateral are perpendicular to each other, the figure would always be included under the general classification: $ \textbf{(A)}\ \text{rhombus} \qquad\textbf{(B)}\ \text{rectangles} \qquad\textbf{(C)}\ \text{square} \qquad\textbf{(D)}\ \text{isosceles trapezoid}\qquad\textbf{(E)}\ \text{none of these} $

Let $ABCD$ be a convex quadrilateral and let $P$ be the intersection of the diagonals $AC$ and $BD$. The radii of the circles inscribed in the triangles $\vartriangle ABP$, $\vartriangle BCP$, $\vartriangle CDP$ and $\vartriangle DAP$ are the same. Prove that $ABCD$ is a rhombus,

Given is a rhombus $ABCD$ of side 1. On the sides $BC$ and $CD$ we are given the points $M$ and $N$ respectively, such that $MC+CN+MN=2$ and $2\angle MAN = \angle BAD$. Find the measures of the angles of the rhombus. [i]Cristinel Mortici[/i]

The rhombus $ABCD$ is given. Let $E$ be one of the points of intersection of the circles $\Gamma_B$ and $\Gamma_C$, where $\Gamma_B$ is the circle centered at $B$ and passing through $C$, and $\Gamma_C$ is the circle centered at $C$ and passing through $B$. The line $ED$ intersects $\Gamma_B$ at point $F$. Find the value of angle $\angle AFB$. [i](S. Mazanik)[/i]

Quadrilateral $ABCD$ is a rhombus with perimeter $52$ meters. The length of diagonal $\overline{AC}$ is $24$ meters. What is the area in square meters of rhombus $ABCD$? [asy] unitsize(1cm); draw((0,1)--(2,2)--(4,1)--(2,0)--cycle); dot("$A$",(0,1),W); dot("$D$",(2,2),N); dot("$C$",(4,1),E); dot("$B$",(2,0),S); [/asy] $\textbf{(A) } 60 \qquad\textbf{(B) } 90 \qquad\textbf{(C) } 105 \qquad\textbf{(D) } 120 \qquad\textbf{(E) } 144$

Let $ABCD$ be a trapezoid with $AB // CD$, $2|AB| = |CD|$ and $BD \perp BC$. Let $M$ be the midpoint of $CD$ and let $E$ be the intersection $BC$ and $AD$. Let $O$ be the intersection of $AM$ and $BD$. Let $N$ be the intersection of $OE$ and $AB$. (a) Prove that $ABMD$ is a rhombus. (b) Prove that the line $DN$ passes through the midpoint of the line segment $BE$.

On the diagonal $AC$ of the rhombus $ABCD$, a point $E$ is taken, which is different from points $A$ and $C$, and on the lines $AB$ and $BC$ are points $N$ and $M$, respectively, with $AE = NE$ and $CE = ME$. Let $K$ be the intersection point of lines $AM$ and $CN$. Prove that points $K, E$ and $D$ are collinear.