Olympiad problems

2020 Iran MO (3rd Round), 1

Tags: geometry , rhombus

Let $ABCD$ be a Rhombus and let $w$ be it's incircle. Let $M$ be the midpoint of $AB$ the point $K$ is on $w$ and inside $ABCD$ such that $MK$ is tangent to $w$. Prove that $CDKM$ is cyclic.

1978 Romania Team Selection Test, 4

Tags: perimeter , rhombus , pure geometry , geometry

Diagonals $ AC $ and $ BD $ of a convex quadrilateral $ ABCD $ intersect a point $ O. $ Prove that if triangles $ OAB,OBC,OCD $ and $ ODA $ have the same perimeter, then $ ABCD $ is a rhombus. What happens if $ O $ is some other point inside the quadrilateral?

2023 Austrian MO Regional Competition, 2

Tags: geometry , concurrency , rhombus

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]

2002 Federal Math Competition of S&M, Problem 3

Tags: inequalities , rhombus , trigonometry , circumcircle , geometric transformation , geometry

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$.

2002 Tournament Of Towns, 5

Tags: trigonometry , rhombus , geometry

Let $AA_1,BB_1,CC_1$ be the altitudes of acute $\Delta ABC$. Let $O_a,O_b,O_c$ be the incentres of $\Delta AB_1C_1,\Delta BC_1A_1,\Delta CA_1B_1$ respectively. Also let $T_a,T_b,T_c$ be the points of tangency of the incircle of $\Delta ABC$ with $BC,CA,AB$ respectively. Prove that $T_aO_cT_bO_aT_cO_b$ is an equilateral hexagon.

2007 IberoAmerican, 6

Tags: geometry , trigonometry , rhombus , combinatorics

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.

1998 Poland - Second Round, 5

Tags: pigeonhole principle , geometry , rhombus , combinatorics

Let $a_1,a_2,\ldots,a_7, b_1,b_2,\ldots,b_7\geq 0$ be real numbers satisfying $a_i+b_i\le 2$ for all $i=\overline{1,7}$. Prove that there exist $k\ne m$ such that $|a_k-a_m|+|b_k-b_m|\le 1$. Thanks for show me the mistake typing

2007 Federal Competition For Advanced Students, Part 2, 3

Tags: rhombus , geometry

Determine all rhombuses $ ABCD$ with the given length $ 2a$ of ist sides by giving the angle $ \alpha \equal{} \angle BAD$, such that there exists a circle which cuts each side of the rhombus in a chord of length $ a$.

2012 District Olympiad, 4

Tags: function , integration , geometry , rhombus , real analysis

Let $f:[0,1]\rightarrow \mathbb{R}$ a differentiable function such that $f(0)=f(1)=0$ and $|f'(x)|\le 1,\ \forall x\in [0,1]$. Prove that: \[\left|\int_0 ^1f(t)dt\right|<\frac{1}{4}\]

1952 Poland - Second Round, 3

Tags: geometry , rectangle , inscribed , square , rhombus

Are the following statements true? a) if the four vertices of a rectangle lie on the four sides of a rhombus, then the sides of the rectangle are parallel to the diagonals of the rhombus; b) if the four vertices of a square lie on the four sides of a rhombus that is not a square, then the sides of the square are parallel to the diagonals of the rhombus.

2008 Hanoi Open Mathematics Competitions, 8

Tags: geometry , rhombus , diagonal

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

2006 IMO Shortlist, 6

Tags: tiling , rhombus , combinatorics , hall s marriage theorem

A holey triangle is an upward equilateral triangle of side length $n$ with $n$ upward unit triangular holes cut out. A diamond is a $60^\circ-120^\circ$ unit rhombus. Prove that a holey triangle $T$ can be tiled with diamonds if and only if the following condition holds: Every upward equilateral triangle of side length $k$ in $T$ contains at most $k$ holes, for $1\leq k\leq n$. [i]Proposed by Federico Ardila, Colombia [/i]

2025 India STEMS Category B, 5

Tags: rhombus , homothety , euler circle , arc midpoint

Let $ABC$ be an acute scalene triangle. Let $D, E$ be points on segments $AB, AC$ respectively, such that $BD=CE$. Prove that the nine-point centers of $ADE$, $ACD$, $ABC$, $AEB$ form a rhombus. [i]Proposed by Malay Mahajan and Siddharth Choppara[/i]

2017 May Olympiad, 3

Tags: geometry , rhombus , area

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 $ABCD$.

2016 Junior Balkan Team Selection Tests - Romania, 4

Tags: geometry , cyclic quadrilateral , angle bisector , rhombus

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 $EF$

2012 USA TSTST, 7

Tags: circumcircle , vector , symmetry , rhombus , gliding principle , geometry

Triangle $ABC$ is inscribed in circle $\Omega$. The interior angle bisector of angle $A$ intersects side $BC$ and $\Omega$ at $D$ and $L$ (other than $A$), respectively. Let $M$ be the midpoint of side $BC$. The circumcircle of triangle $ADM$ intersects sides $AB$ and $AC$ again at $Q$ and $P$ (other than $A$), respectively. Let $N$ be the midpoint of segment $PQ$, and let $H$ be the foot of the perpendicular from $L$ to line $ND$. Prove that line $ML$ is tangent to the circumcircle of triangle $HMN$.

2015 Auckland Mathematical Olympiad, 2

Tags: rhombus , perimeter , geometry

A convex quadrillateral $ABCD$ is given and the intersection point of the diagonals is denoted by $O$. Given that the perimeters of the triangles $ABO, BCO, CDO,ADO$ are equal, prove that $ABCD$ is a rhombus.

2011 AMC 10, 20

Tags: rhombus , trigonometry , perpendicular bisector , geometry

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 $

2003 Brazil National Olympiad, 3

Tags: geometry , rhombus , trigonometry , projective geometry

$ABCD$ 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.

2003 Romania National Olympiad, 1

Tags: geometry , rhombus , analytic geometry , pure geometry

Find the locus of the points $ M $ that are situated on the plane where a rhombus $ ABCD $ lies, and satisfy: $$ MA\cdot MC+MB\cdot MD=AB^2 $$ [i]Ovidiu Pop[/i]

2018 Yasinsky Geometry Olympiad, 5

Tags: geometry , angle chasing , rhombus

The point $M$ lies inside the rhombus $ABCD$. It is known that $\angle DAB=110^o$, $\angle AMD=80^o$, $\angle BMC= 100^o$. What can the angle $\angle AMB$ be equal?

2025 Portugal MO, 4

Tags: geometry , rhombus

Let $ABCD$ be a square with $2cm$ side length and with center $T$. A rhombus $ARTE$ is drawn where point $E$ lies on line $DC$. What is the area of $ARTE$?

2013 ITAMO, 5

Tags: circumcircle , parallelogram , rhombus , perpendicular bisector , geometry

$ABC$ is an isosceles triangle with $AB=AC$ and the angle in $A$ is less than $60^{\circ}$. Let $D$ be a point on $AC$ such that $\angle{DBC}=\angle{BAC}$. $E$ is the intersection between the perpendicular bisector of $BD$ and the line parallel to $BC$ passing through $A$. $F$ is a point on the line $AC$ such that $FA=2AC$ ($A$ is between $F$ and $C$). Show that $EB$ and $AC$ are parallel and that the perpendicular from $F$ to $AB$, the perpendicular from $E$ to $AC$ and $BD$ are concurrent.

1994 National High School Mathematics League, 6

Tags: geometry , rectangle , rhombus

In rectangular coordinate system, the equation $\frac{|x+y|}{2a}+\frac{|x-y|}{2b}=1$ ($a,b$ are different positive numbers) refers to $\text{(A)}$ a triangle $\text{(B)}$ a square $\text{(C)}$ rectangle, not square $\text{(D)}$ rhombus, not square

2007 Iran MO (3rd Round), 1

Tags: polynomial , geometry , rhombus , algebra

Let $ a,b$ be two complex numbers. Prove that roots of $ z^{4}\plus{}az^{2}\plus{}b$ form a rhombus with origin as center, if and only if $ \frac{a^{2}}{b}$ is a non-positive real number.