Olympiad problems

Kvant 2020, M2622

Tags: geometry , rhombus

The points $E, F, G$ and $H{}$ are located on the sides $DA, AB, BC$ and $CD$ of the rhombus $ABCD$ respectively, so that the segments $EF$ and $GH$ touch the circle inscribed in the rhombus. Prove that $FG\parallel HE$. [i]Proposed by V. Eisenstadt[/i]

2008 Thailand Mathematical Olympiad, 1

Tags: geometry , rhombus , tangent

Let $P$ be a point outside a circle $\omega$. The tangents from $P$ to $\omega$ are drawn touching $\omega$ at points $A$ and $B$. Let $M$ and $N$ be the midpoints of $AP$ and $AB$, respectively. Line $MN$ is extended to cut $\omega$ at $C$ so that $N$ lies between $M$ and $C$. Line $PC$ intersects $\omega$ again at $D$, and lines $ND$ and $PB$ intersect at $O$. Prove that $MNOP$ is a rhombus.

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

Ukrainian From Tasks to Tasks - geometry, 2013.9

Tags: geometry , rhombus , trisection

The perpendicular bisectors of the sides $AB$ and $CD$ of the rhombus $ABCD$ are drawn. It turned out that they divided the diagonal $AC$ into three equal parts. Find the altitude of the rhombus if $AB = 1$.

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]

1997 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: geometry , 3-dimensional geometry , prism , sphere , rhombus , 3d geometry

Consider a prism, not necessarily right, whose base is a rhombus $ABCD$ with side $AB = 5$ and diagonal $AC = 8$. A sphere of radius $r$ is tangent to the plane $ABCD$ at $C$ and tangent to the edges $AA_1$ , $BB _1$ and $DD_ 1$ of the prism. Calculate $r$ .

2010 All-Russian Olympiad, 4

Tags: rhombus , geometry

In a acute triangle $ABC$, the median, $AM$, is longer than side $AB$. Prove that you can cut triangle $ABC$ into $3$ parts out of which you can construct a rhombus.

2017 OMMock - Mexico National Olympiad Mock Exam, 1

Tags: geometry , circumcenter , rhombus , circumcircle

Let $ABC$ be a triangle with circumcenter $O$. Point $D, E, F$ are chosen on sides $AB, BC$ and $AC$, respectively, such that $ADEF$ is a rhombus. The circumcircles of $BDE$ and $CFE$ intersect $AE$ at $P$ and $Q$ respectively. Show that $OP=OQ$. [i]Proposed by Ariel García[/i]

1979 IMO Shortlist, 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).

2023 Ukraine National Mathematical Olympiad, 8.4

Tags: geometry , rhombus , tangency

Point $T$ is chosen in the plane of a rhombus $ABCD$ so that $\angle ATC + \angle BTD = 180^\circ$, and circumcircles of triangles $ATC$ and $BTD$ are tangent to each other. Show that $T$ is equidistant from diagonals of $ABCD$. [i]Proposed by Fedir Yudin[/i]

2008 Baltic Way, 16

Tags: parallelogram , rhombus , geometry

Let $ABCD$ be a parallelogram. The circle with diameter $AC$ intersects the line $BD$ at points $P$ and $Q$. The perpendicular to the line $AC$ passing through the point $C$ intersects the lines $AB$ and $AD$ at points $X$ and $Y$, respectively. Prove that the points $P,Q,X$ and $Y$ lie on the same circle.

2001 Baltic Way, 9

Tags: rhombus , parallelogram , circumcircle , hyperbola , cyclic quadrilateral , conic , geometry

Given a rhombus $ABCD$, find the locus of the points $P$ lying inside the rhombus and satisfying $\angle APD+\angle BPC=180^{\circ}$.

2010 Contests, 1

Tags: ratio , circumcircle , parallelogram , incenter , rhombus , geometry

$ABC$ is an acute angle triangle such that $AB>AC$ and $\hat{BAC}=60^{\circ}$. Let's denote by $O$ the center of the circumscribed circle of the triangle and $H$ the intersection of altitudes of this triangle. Line $OH$ intersects $AB$ in point $P$ and $AC$ in point $Q$. Find the value of the ration $\frac{PO}{HQ}$.

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.

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 $ARTE$?

2000 Singapore Team Selection Test, 2

Tags: geometry , rhombus

In a triangle $ABC$, $\angle C = 60^o$, $D, E, F$ are points on the sides $BC, AB, AC$ respectively, and $M$ is the intersection point of $AD$ and $BF$. Suppose that $CDEF$ is a rhombus. Prove that $DF^2 = DM \cdot DA$

1967 All Soviet Union Mathematical Olympiad, 092

Tags: geometry , rhombus

Three vertices $KLM$ of the rhombus (diamond) $KLMN$ lays on the sides $[AB], [BC]$ and $[CD]$ of the given unit square. Find the area of the set of all the possible vertices $N$.

2018-2019 SDML (High School), 8

Tags: geometry , rhombus

The figure below consists of five isosceles triangles and ten rhombi. The bases of the isosceles triangles are $12$, $13$, $14$, $15$, as shown below. The top rhombus, which is shaded, is actually a square. Find the area of this square. [DIAGRAM NEEDED]

2010 Dutch BxMO TST, 1

Tags: geometry , rhombus , trapezoid , midpoint

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 $BE$.

2011 Turkey Team Selection Test, 2

Tags: geometry , incenter , circumcircle , geometric transformation , reflection , rhombus , symmetry

Let $I$ be the incenter and $AD$ be a diameter of the circumcircle of a triangle $ABC.$ If the point $E$ on the ray $BA$ and the point $F$ on the ray $CA$ satisfy the condition \[BE=CF=\frac{AB+BC+CA}{2}\] show that the lines $EF$ and $DI$ are perpendicular.

2012 NIMO Problems, 7

Tags: trapezoid , rhombus , geometry

In quadrilateral $ABCD$, $AC = BD$ and $\measuredangle B = 60^\circ$. Denote by $M$ and $N$ the midpoints of $\overline{AB}$ and $\overline{CD}$, respectively. If $MN = 12$ and the area of quadrilateral $ABCD$ is 420, then compute $AC$. [i]Proposed by Aaron Lin[/i]

2010 Dutch BxMO TST, 1

Tags: geometry , rhombus , trapezoid , midpoint

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 $BE$.

2004 Austria Beginners' Competition, 4

Tags: geometry , rhombus , construction

Of a rhombus $ABCD$ we know the circumradius $R$ of $\Delta ABC$ and $r$ of $\Delta BCD$. Construct the rhombus.

2014 Brazil National Olympiad, 1

Tags: geometry , incenter , trigonometry , rhombus , inradius

Let $ABCD$ be a convex quadrilateral. Diagonals $AC$ and $BD$ meet at point $P$. The inradii of triangles $ABP$, $BCP$, $CDP$ and $DAP$ are equal. Prove that $ABCD$ is a rhombus.

2012 All-Russian Olympiad, 4

Tags: incenter , geometric transformation , homothety , reflection , trigonometry , rhombus , geometry

The point $E$ is the midpoint of the segment connecting the orthocentre of the scalene triangle $ABC$ and the point $A$. The incircle of triangle $ABC$ incircle is tangent to $AB$ and $AC$ at points $C'$ and $B'$ respectively. Prove that point $F$, the point symmetric to point $E$ with respect to line $B'C'$, lies on the line that passes through both the circumcentre and the incentre of triangle $ABC$.