Found problems: 264
2004 Brazil Team Selection Test, Problem 3
Determine the locus of points $M$ in the plane of a given rhombus $ABCD$ such that $MA\cdot MC+MB\cdot MD=AB^2$.
1995 National High School Mathematics League, 3
Inscribed Circle of rhombus $ABCD$ touches $AB,BC,CD,DA$ at $E,F,G,H$. $l_1,l_2$ are two lines that are tangent to the circle. $l_1\cap AB=M,l_1\cap BC=N,l_2\cap CD=P,l_2\cap DA=Q$. Prove that $MQ/\! /NP$.
2006 Australia National Olympiad, 1
In a square $ABCD$, $E$ is a point on diagonal $BD$. $P$ and $Q$ are the circumcentres of $\triangle ABE$ and $\triangle ADE$ respectively. Prove that $APEQ$ is a square.
1999 AMC 12/AHSME, 16
What is the radius of a circle inscribed in a rhombus with diagonals of length $ 10$ and $ 24$?
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 58/13 \qquad
\textbf{(C)}\ 60/13 \qquad
\textbf{(D)}\ 5 \qquad
\textbf{(E)}\ 6$
1953 AMC 12/AHSME, 15
A circular piece of metal of maximum size is cut out of a square piece and then a square piece of maximum size is cut out of the circular piece. The total amount of metal wasted is:
$ \textbf{(A)}\ \frac{1}{4} \text{ the area of the original square}\\
\textbf{(B)}\ \frac{1}{2} \text{ the area of the original square}\\
\textbf{(C)}\ \frac{1}{2} \text{ the area of the circular piece}\\
\textbf{(D)}\ \frac{1}{4} \text{ the area of the circular piece}\\
\textbf{(E)}\ \text{none of these}$
2018 Stars of Mathematics, 3
Let be an isosceles trapezoid such that its smaller base is equal to its legs, and a rhombus that has each of its vertexes on a different side of the trapezoid. Prove that the smaller angles of the trapezoid are equal to the smaller ones of the rhombus.
[i]Vlad Robu[/i]
2002 BAMO, 2
In the illustration, a regular hexagon and a regular octagon have been tiled with rhombuses.
In each case, the sides of the rhombuses are the same length as the sides of the regular polygon.
(a) Tile a regular decagon ($10$-gon) into rhombuses in this manner.
(b) Tile a regular dodecagon ($12$-gon) into rhombuses in this manner.
(c) How many rhombuses are in a tiling by rhombuses of a $2002$-gon?
Justify your answer.
[img]https://cdn.artofproblemsolving.com/attachments/8/a/8413e4e2712609eba07786e34ba2ce4aa72888.png[/img]
2014 Contests, 2
Outside the square $ABCD$, the rhombus $BCMN$ is constructed with angle $BCM$ obtuse . Let $P$ be the intersection point of the lines $BM$ and $AN$ . Prove that $DM \perp CP$ and the triangle $DPM$ is right isosceles .
1975 Kurschak Competition, 2
Prove or disprove: given any quadrilateral inscribed in a convex polygon, we can find a rhombus inscribed in the polygon with side not less than the shortest side of the quadrilateral.
1954 AMC 12/AHSME, 45
In a rhombus, $ ABCD$, line segments are drawn within the rhombus, parallel to diagonal $ BD$, and terminated in the sides of the rhombus. A graph is drawn showing the length of a segment as a function of its distance from vertex $ A$. The graph is:
$ \textbf{(A)}\ \text{A straight line passing through the origin.} \\
\textbf{(B)}\ \text{A straight line cutting across the upper right quadrant.} \\
\textbf{(C)}\ \text{Two line segments forming an upright V.} \\
\textbf{(D)}\ \text{Two line segments forming an inverted V.} \\
\textbf{(E)}\ \text{None of these.}$
2023 Regional Competition For Advanced Students, 2
Let $ABCD$ be a rhombus with $\angle BAD < 90^o$. The circle passing through $D$ with center $A$ intersects the line $CD$ a second time in point $E$. Let $S$ be the intersection of the lines $BE$ and $AC$. Prove that the points $A$, $S$, $D$ and $E$ lie on a circle.
[i](Karl Czakler)[/i]
2019 Latvia Baltic Way TST, 9
Let $ABCD$ be a rhombus with the condition $\angle ABC > 90^o$. The circle $\Gamma_B$ with center at $B$ goes through $C$, and the circle $\Gamma_C$ with center at $C$ goes through $B$. Denote by $E$ one of the intersection points of $\Gamma_B$ and $\Gamma_C$. The line $ED$ intersects intersects $\Gamma_B$ again at $F$. Find the value of $\angle AFB$.
1999 All-Russian Olympiad, 3
A circle touches sides $DA$, $AB$, $BC$, $CD$ of a quadrilateral $ABCD$ at points $K$, $L$, $M$, $N$, respectively. Let $S_1$, $S_2$, $S_3$, $S_4$ respectively be the incircles of triangles $AKL$, $BLM$, $CMN$, $DNK$. The external common tangents distinct from the sides of $ABCD$ are drawn to $S_1$ and $S_2$, $S_2$ and $S_3$, $S_3$ and $S_4$, $S_4$ and $S_1$. Prove that these four tangents determine a rhombus.
2009 Junior Balkan Team Selection Tests - Romania, 2
Consider a rhombus $ABCD$. Point $M$ and $N$ are given on the line segments $AC$ and $BC$ respectively, such that $DM = MN$. Lines $AC$ and $DN$ meet at point $P$ and lines $AB$ and $DM$ meet at point $R$. Prove that $RP = PD$.
2017 Azerbaijan Junior National Olympiad, P4
A Rhombus and an Isosceles trapezoid that has same area is drawn in the same circle's outside. Compare their acute angles \\
(explain your answer)
2013 Iran MO (3rd Round), 4
We have constructed a rhombus by attaching two equal equilateral triangles. By putting $n-1$ points on all 3 sides of each triangle we have divided the sides to $n$ equal segments. By drawing line segements between correspounding points on each side of the triangles we have divided the rhombus into $2n^2$ equal triangles.
We write the numbers $1,2,\dots,2n^2$ on these triangles in a way no number appears twice. On the common segment of each two triangles we write the positive difference of the numbers written on those triangles. Find the maximum sum of all numbers written on the segments.
(25 points)
[i]Proposed by Amirali Moinfar[/i]
2017 May Olympiad, 3
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 $ABCD$.
2020 Canadian Junior Mathematical Olympiad, 4
$ABCD$ is a fixed rhombus. Segment $PQ$ is tangent to the inscribed circle of $ABCD$, where $P$ is on side $AB$, $Q$ is on side $AD$. Show that, when segment $PQ$ is moving, the area of $\Delta CPQ$ is a constant.
1999 Estonia National Olympiad, 4
We build rhombuses from natural numbers. Find the sum of the numbers in the $n$-th rhombus.
[img]https://cdn.artofproblemsolving.com/attachments/e/7/22360573f76c615ca43bbacb8f15e587772ca4.png[/img]
Brazil L2 Finals (OBM) - geometry, 2001.1
A sheet of rectangular $ABCD$ paper, of area $1$, is folded along its diagonal $AC$ and then unfolded, then it is bent so that vertex $A$ coincides with vertex $C$ and then unfolded, leaving the crease $MN$, as shown below.
a) Show that the quadrilateral $AMCN$ is a rhombus.
b) If the diagonal $AC$ is twice the width $AD$, what is the area of the rhombus $AMCN$?
[img]https://2.bp.blogspot.com/-TeQ0QKYGzOQ/Xp9lQcaLbsI/AAAAAAAAL2E/JLXwEIPSr4U79tATcYzmcJjK5bGA6_RqACK4BGAYYCw/s400/2001%2Baomb%2Bl2.png[/img]
1992 IberoAmerican, 2
Given a circle $\Gamma$ and the positive numbers $h$ and $m$, construct with straight edge and compass a trapezoid inscribed in $\Gamma$, such that it has altitude $h$ and the sum of its parallel sides is $m$.
2018 Hanoi Open Mathematics Competitions, 8
Let $ABCD$ be rhombus, with $\angle ABC = 80^o$: Let $E$ be midpoint of $BC$ and $F$ be perpendicular projection of $A$ onto $DE$. Find the measure of $\angle DFC$ in degree.
2008 Swedish Mathematical Competition, 1
A rhombus is inscribed in a convex quadrilateral. The sides of the rhombus are parallel with the diagonals of the quadrilateral, which have the lengths $d_1$ and $d_2$. Calculate the length of side of the rhombus , expressed in terms of $d_1$ and $d_2$.
2008 National Olympiad First Round, 33
Let $E$ be a point inside the rhombus $ABCD$ such that $|AE|=|EB|$, $m(\widehat{EAB})=12^\circ$, and $m(\widehat{DAE})=72^\circ$. What is $m(\widehat{CDE})$ in degrees?
$
\textbf{(A)}\ 64
\qquad\textbf{(B)}\ 66
\qquad\textbf{(C)}\ 68
\qquad\textbf{(D)}\ 70
\qquad\textbf{(E)}\ 72
$
1961 AMC 12/AHSME, 14
A rhombus is given with one diagonal twice the length of the other diagonal. Express the side of the rhombus is terms of $K$, where $K$ is the area of the rhombus in square inches.
${{ \textbf{(A)}\ \sqrt{K} \qquad\textbf{(B)}\ \frac{1}{2}\sqrt{2K} \qquad\textbf{(C)}\ \frac{1}{3}\sqrt{3K} \qquad\textbf{(D)}\ \frac{1}{4}\sqrt{4K} }\qquad\textbf{(E)}\ \text{None of these are correct} } $