Olympiad problems

May Olympiad L1 - geometry, 2017.3

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 $ABCD$.

2021 Taiwan TST Round 3, 6

Tags: rhombus , incenter , geometry

Let $ ABCD $ be a rhombus with center $ O. $ $ P $ is a point lying on the side $ AB. $ Let $ I, $ $ J, $ and $ L $ be the incenters of triangles $ PCD, $ $ PAD, $ and $PBC, $ respectively. Let $ H $ and $ K $ be orthocenters of triangles $ PLB $ and $ PJA, $ respectively. Prove that $ OI \perp HK. $ [i]Proposed by buratinogigle[/i]

2000 Turkey Team Selection Test, 2

Tags: rhombus , circumcircle , trigonometry , angle bisector , geometry

Points $M,\ N,\ K,\ L$ are taken on the sides $AB,\ BC,\ CD,\ DA$ of a rhombus $ABCD,$ respectively, in such a way that $MN\parallel LK$ and the distance between $MN$ and $KL$ is equal to the height of $ABCD.$ Show that the circumcircles of the triangles $ALM$ and $NCK$ intersect each other, while those of $LDK$ and $MBN$ do not.

2010 Bosnia and Herzegovina Junior BMO TST, 3

Tags: equal area , rhombus , geometry

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

Kyiv City MO Juniors 2003+ geometry, 2013.8.5

Tags: geometry , rhombus , circles , concurrency

Let $ABCD$ be a convex quadrilateral. Prove that the circles inscribed in the triangles $ABC$, $BCD$, $CDA$ and $DAB$ have a common point if and only if $ABCD$ is a rhombus.

2005 Poland - Second Round, 2

Tags: rhombus , trapezoid , geometry

A rhombus $ABCD$ with $\angle BAD=60^{\circ}$ is given. Points $E$ on side $AB$ and $F$ on side $AD$ are such that $\angle ECF=\angle ABD$. Lines $CE$ and $CF$ respectively meet line $BD$ at $P$ and $Q$. Prove that $\frac{PQ}{EF}=\frac{AB}{BD}$.

1979 IMO Longlists, 1

Tags: geometry , polygon , dissection , rhombus

Prove that in the Euclidean plane every regular polygon having an even number of sides can be dissected into lozenges. (A lozenge is a quadrilateral whose four sides are all of equal length).

1906 Eotvos Mathematical Competition, 2

Tags: geometry , rhombus , square

Let $K, L,M,N$ designate the centers of the squares erected on the four sides (outside) of a rhombus. Prove that the polygon $KLMN$ is a square.

2022 AMC 10, 20

Tags: geometry , rhombus , solvable without cyclic

Let $ABCD$ be a rhombus with $\angle{ADC} = 46^{\circ}$. Let $E$ be the midpoint of $\overline{CD}$, and let $F$ be the point on $\overline{BE}$ such that $\overline{AF}$ is perpendicular to $\overline{BE}$. What is the degree measure of $\angle{BFC}$? $\textbf{(A)}\ 110 \qquad \textbf{(B)}\ 111 \qquad \textbf{(C)}\ 112 \qquad \textbf{(D)}\ 113 \qquad \textbf{(E)}\ 114$

2014 Germany Team Selection Test, 3

Tags: geometry , rhombus , trigonometry

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 $\varphi \le \max \{ \angle BAC, \angle ABC \}$.

2025 Israel National Olympiad (Gillis), P2

Tags: square , geometry , rhombus , collinearity

Let $ABCD$ be a rhombus. Eight additional points $X_1$, $X_2$, $Y_1$, $Y_2$, $Z_1$, $Z_2$, $W_1$, $W_2$ were chosen so that the quadrilaterals $AX_1BX_2$, $BY_1CY_2$, $CZ_1DZ_2$, $DW_1AW_2$ are squares. Prove that the eight new points lie on two straight lines.

2011 AMC 12/AHSME, 16

Tags: geometry , rhombus , trigonometry , perpendicular bisector

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 $

2004 Brazil Team Selection Test, Problem 3

Tags: rhombus , geometry

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 Romania Team Selection Test, 4

Tags: geometry , rhombus , square , isosceles

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.

2013 Stanford Mathematics Tournament, 5

Tags: rhombus , perimeter , geometry

A rhombus has area $36$ and the longer diagonal is twice as long as the shorter diagonal. What is the perimeter of the rhombus?

1996 APMO, 1

Tags: perimeter , ratio , rhombus , trigonometry , invariant , symmetry , geometry

Let $ABCD$ be a quadrilateral $AB = BC = CD = DA$. Let $MN$ and $PQ$ be two segments perpendicular to the diagonal $BD$ and such that the distance between them is $d > \frac{BD}{2}$, with $M \in AD$, $N \in DC$, $P \in AB$, and $Q \in BC$. Show that the perimeter of hexagon $AMNCQP$ does not depend on the position of $MN$ and $PQ$ so long as the distance between them remains constant.

1988 AMC 12/AHSME, 6

Tags: geometry , parallelogram , rectangle , rhombus , trapezoid

A figure is an equiangular parallelogram if and only if it is a $ \textbf{(A)}\ \text{rectangle}\qquad\textbf{(B)}\ \text{regular polygon}\qquad\textbf{(C)}\ \text{rhombus}\qquad\textbf{(D)}\ \text{square}\qquad\textbf{(E)}\ \text{trapezoid} $

2005 Taiwan TST Round 3, 2

Tags: parallelogram , rhombus , geometry

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.

1989 IMO Shortlist, 32

Tags: circumcircle , trigonometry , parallelogram , rhombus , angle , geometry

The vertex $ A$ of the acute triangle $ ABC$ is equidistant from the circumcenter $ O$ and the orthocenter $ H.$ Determine all possible values for the measure of angle $ A.$

1951 Polish MO Finals, 5

Tags: rhombus , geometry , cyclic quadrilateral

A quadrilateral $ ABCD $ is inscribed in a circle. The lines $AB$ and $CD$ intersect at point $E$, and the lines $AD$ and $BC$ intersect at point $F$. The bisector of the angle $ AEC $ intersects the side $ BC $ at the point $ M $ and the side $ AD $ at the point $ N $; and the bisector of the angle $ BFD $ intersects the side $ AB $ at the point $ P $ and the side $ CD $ at the point $ Q $. Prove that the quadrilateral $MPNQ$ is a rhombus.

2002 National Olympiad First Round, 33

Tags: geometry , rhombus

Let $ABCD$ be a rhombus such that $m(\widehat{ABC}) = 40^\circ$. Let $E$ be the midpoint of $[BC]$ and $F$ be the foot of the perpendicular from $A$ to $DE$. What is $m(\widehat{DFC})$? $ \textbf{a)}\ 100^\circ \qquad\textbf{b)}\ 110^\circ \qquad\textbf{c)}\ 115^\circ \qquad\textbf{d)}\ 120^\circ \qquad\textbf{e)}\ 135^\circ $

2012 NIMO Problems, 6

Tags: geometry , rhombus , perimeter , trigonometry , circumcircle , trig identities

In rhombus $NIMO$, $MN = 150\sqrt{3}$ and $\measuredangle MON = 60^{\circ}$. Denote by $S$ the locus of points $P$ in the interior of $NIMO$ such that $\angle MPO \cong \angle NPO$. Find the greatest integer not exceeding the perimeter of $S$. [i]Proposed by Evan Chen[/i]

1991 IMO Shortlist, 7

Tags: geometry , 3d geometry , tetrahedron , rhombus

$ ABCD$ is a terahedron: $ AD\plus{}BD\equal{}AC\plus{}BC,$ $ BD\plus{}CD\equal{}BA\plus{}CA,$ $ CD\plus{}AD\equal{}CB\plus{}AB,$ $ M,N,P$ are the mid points of $ BC,CA,AB.$ $ OA\equal{}OB\equal{}OC\equal{}OD.$ Prove that $ \angle MOP \equal{} \angle NOP \equal{}\angle NOM.$

2003 Romania Team Selection Test, 14

Tags: rhombus , geometry

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]

2007 AMC 12/AHSME, 19

Tags: geometry , rhombus , trigonometry

Rhombus $ ABCD$, with a side length $ 6$, is rolled to form a cylinder of volume $ 6$ by taping $ \overline{AB}$ to $ \overline{DC}.$ What is $ \sin(\angle ABC)$? $ \textbf{(A)}\ \frac {\pi}{9} \qquad \textbf{(B)}\ \frac {1}{2} \qquad \textbf{(C)}\ \frac {\pi}{6} \qquad \textbf{(D)}\ \frac {\pi}{4} \qquad \textbf{(E)}\ \frac {\sqrt3}{2}$