Let $ ABCD$ be a rhombus with $ AC\equal{}16$ and $ BD\equal{}30$. Let $ N$ be a point on $ \overline{AB}$, and let $ P$ and $ Q$ be the feet of the perpendiculars from $ N$ to $ \overline{AC}$ and $ \overline{BD}$, respectively. Which of the following is closest to the minimum possible value of $ PQ$? [asy]unitsize(2.5cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair D=(0,0), C=dir(0), A=dir(aSin(240/289)), B=shift(A)*C; pair Np=waypoint(B--A,0.6), P=foot(Np,A,C), Q=foot(Np,B,D); draw(A--B--C--D--cycle); draw(A--C); draw(B--D); draw(Np--Q); draw(Np--P); label("$D$",D,SW); label("$C$",C,SE); label("$B$",B,NE); label("$A$",A,NW); label("$N$",Np,N); label("$P$",P,SW); label("$Q$",Q,SSE); draw(rightanglemark(Np,P,C,2)); draw(rightanglemark(Np,Q,D,2));[/asy]$ \textbf{(A)}\ 6.5 \qquad \textbf{(B)}\ 6.75 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 7.25 \qquad \textbf{(E)}\ 7.5$

Let $ABCD$ be a convex quadrilateral. Suppose that similar isosceles triangles $APB, BQC, CRD, DSA$ with the bases on the sides of $ABCD$ are constructed in the exterior of the quadrilateral such that $PQRS$ is a rectangle but not a square. Show that $ABCD$ is a rhombus.

Let $ \mathcal{F}$ be a family of hexagons $ H$ satisfying the following properties: i) $ H$ has parallel opposite sides. ii) Any 3 vertices of $ H$ can be covered with a strip of width 1. Determine the least $ \ell\in\mathbb{R}$ such that every hexagon belonging to $ \mathcal{F}$ can be covered with a strip of width $ \ell$. Note: A strip is the area bounded by two parallel lines separated by a distance $ \ell$. The lines belong to the strip, too.

Let $n>3$ be a positive integer. Equilateral triangle ABC is divided into $n^2$ smaller congruent equilateral triangles (with sides parallel to its sides). Let $m$ be the number of rhombuses that contain two small equilateral triangles and $d$ the number of rhombuses that contain eight small equilateral triangles. Find the difference $m-d$ in terms of $n$.

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.

The sides of a rhombus have length $a$ and the area is $S$. What is the length of the shorter diagonal?

A circle is inscribed in the trapezoid [i]ABCD[/i]. Let [i]K, L, M, N[/i] be the points of tangency of this circle with the diagonals [i]AC[/i] and [i]BD[/i], respectively ([i]K[/i] is between [i]A[/i] and [i]L[/i], and [i]M[/i] is between [i]B[/i] and [i]N[/i]). Given that $AK\cdot LC=16$ and $BM\cdot ND=\frac94$, find the radius of the circle. [color=red][Moderator edit: A solution of this problem can be found on http://www.ajorza.org/math/mathfiles/scans/belarus.pdf , page 20 (the statement of the problem is on page 6). The author of the problem is I. Voronovich.][/color]

In a parallelogram $\Box ABCD$ $(AB < BC)$ The incircle of $\triangle ABC$ meets $\overline {BC}$ and $\overline {CA}$ at $P, Q$. The incircle of $\triangle ACD$ and $\overline {CD}$ meets at $R$. Let $S$ = $PQ$ $\cap$ $AD$ $U$ = $AR$ $\cap$ $CS$ $T$, a point on $\overline {BC}$ such that $\overline {AB} = \overline {BT}$ Prove that $AT, BU, PQ$ are concurrent

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]

Rhombus $ABCD$ has side length $2$ and $\angle B = 120 ^\circ$. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$? $ \textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad \textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad \textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad \textbf{(D)}\ 1+\frac{\sqrt{3}}{3} \qquad \textbf{(E)}\ 2 $

Convex quadrilateral $ABCD$ with $AB$ not equal to $DC$ is inscribed in a circle. Let $AKDL$ and $CMBN$ be rhombs with same side of $a$. Prove that the points $K, L, M, N$ lie on a circle.

Points $M$ and $N$ are given on sides $AD$ and $BC$ of rhombus $ABCD$, respectively. Line $MC$ intersects line $BD$ in point $T$, line $MN$ intersects line $BD$ in point $U$, line $CU$ intersects line $AB$ in point $Q$ and line $QT$ intersects line $CD$ in $P$. Prove that triangles $QCP$ and $MCN$ have equal area

Let $AB\varGamma\varDelta$ be a rhombus. (a) Prove that you can construct a circle $(c)$ which is inscribed in the rhombus and is tangent to its sides. (b) The points $\varTheta,H,K,I$ are on the sides $\varDelta\varGamma,B\varGamma,AB,A\varDelta$ of the rhombus respectively, such that the line segments $KH$ and $I\varTheta$ are tangent on the circle $(c)$. Prove that the quadrilateral defined from the points $\varTheta,H,K,I$ is a trapezium.

$ABCD$ is a regular tetrahedron with side length $1$. Find the area of the cross section of $ABCD$ cut by the plane that passes through the midpoints of $AB$, $AC$, and $CD$.

Each angle of a rectangle is trisected. The intersections of the pairs of trisectors adjacent to the same side always form: $ \textbf{(A)}\ \text{a square} \qquad\textbf{(B)}\ \text{a rectangle} \qquad\textbf{(C)}\ \text{a parallelogram with unequal sides} \\ \textbf{(D)}\ \text{a rhombus} \qquad\textbf{(E)}\ \text{a quadrilateral with no special properties}$

Let $ABCD$ be a convex quadrilateral with $\angle ABC = 120^{\circ}$ and $\angle BCD = 90^{\circ}$, and let $M$ and $N$ denote the midpoints of $\overline{BC}$ and $\overline{CD}$. Suppose there exists a point $P$ on the circumcircle of $\triangle CMN$ such that ray $MP$ bisects $\overline{AD}$ and ray $NP$ bisects $\overline{AB}$. If $AB + BC = 444$, $CD = 256$ and $BC = \frac mn$ for some relatively prime positive integers $m$ and $n$, compute $100m+n$. [i]Proposed by Michael Ren[/i]

The figure shows a parallelogram and the point $P$ of intersection of its diagonals is marked. Draw a straight line through $P$ so that it breaks the parallelogram into two parts, from which you can fold a rhombus. [img]https://1.bp.blogspot.com/-Df2tIBthcmI/X2ZwIx3R4vI/AAAAAAAAMhQ/8Zkxfq30H8MSCdc66tm33n6jt-QKfGMowCLcBGAsYHQ/s0/2009%2Boral%2Bmoscow%2Bj1.png[/img]

We have a rhombus $ABCD$ with $\angle BAD=60^\circ$. We take points $F,H,G$ on the sides $AD,DC$ and the diagonal $AC$, respectively, such that $DFGH$ is a parallelogram. Prove that $BFH$ is equilateral.

Rhombus $ABCD$ has side length $2$ and $\angle B = 120 ^\circ$. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$? $ \textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad \textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad \textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad \textbf{(D)}\ 1+\frac{\sqrt{3}}{3} \qquad \textbf{(E)}\ 2 $

Olga and Sasha play a game on an infinite hexagonal grid. They take turns in placing a stone on a free hexagon of their choice. Olga starts the game. Just before the $2018$th stone is placed, a new rule comes into play. A stone may now be placed only on those free hexagons having at least two occupied neighbors. A player loses when she or he either is unable to make a move, or makes a move such that a pattern of the rhomboid shape as shown (rotated in any possible way) appears. Determine which player, if any, possesses a winning strategy.

Let $ABCD$ be a rhombus where $\angle DAB = 60^\circ$, and $P$ be the intersection between $AC$ and $BD$. Let $Q,R,S$ be three points on the boundary of $ABCD$ such that $PQRS$ is a rhombus. Prove that exactly one of $Q,R,S$ lies on one of $A,B,C,D$.

Let $n>3$ be a positive integer. Equilateral triangle ABC is divided into $n^2$ smaller congruent equilateral triangles (with sides parallel to its sides). Let $m$ be the number of rhombuses that contain two small equilateral triangles and $d$ the number of rhombuses that contain eight small equilateral triangles. Find the difference $m-d$ in terms of $n$.

In the Cartesian plane $XOY$, there is a rhombus $ABCD$ whose side lengths are all $4$ and $|OB|=|OD|=6$, where $O$ is the origin. [b](1)[/b] Prove that $|OA|\cdot |OB|$ is a constant. [b](2)[/b] Find the locus of $C$ if $A$ is a point on the semicircle \[(x-2)^2+y^2=4 \quad (2\le x\le 4).\]

A rhombus is formed by two radii and two chords of a circle whose radius is $ 16$ feet. The area of the rhombus in square feet is: $ \textbf{(A)}\ 128 \qquad\textbf{(B)}\ 128\sqrt {3} \qquad\textbf{(C)}\ 256 \qquad\textbf{(D)}\ 512 \qquad\textbf{(E)}\ 512\sqrt {3}$

$ABCD$ is a rhombus. Take points $E$, $F$, $G$, $H$ on sides $AB$, $BC$, $CD$, $DA$ respectively so that $EF$ and $GH$ are tangent to the incircle of $ABCD$. Show that $EH$ and $FG$ are parallel.