Olympiad problems

2014 Contests, 3

Let $r,R$ and $r_a$ be the radii of the incircle, circumcircle and A-excircle of the triangle $ABC$ with $AC>AB$, respectively. $I,O$ and $J_A$ are the centers of these circles, respectively. Let incircle touches the $BC$ at $D$, for a point $E \in (BD)$ the condition $A(IEJ_A)=2A(IEO)$ holds. Prove that \[ED=AC-AB \iff R=2r+r_a.\]

2016 Portugal MO, 4

Tags: geometry , equal angles , tangent , angle , parallelogram

Let $[ABCD]$ be a parallelogram with $AB <BC$ and let $E, F$ be points on the circle that passes through $A, B$ and $C$ such that $DE$ and $DF$ are tangents to this circle. Knowing that $\angle ADE = \angle CDF$ , determine $\angle ABC$. [img]https://cdn.artofproblemsolving.com/attachments/5/e/4140b92730e9d382df49ac05ca4e8ba48332dc.png[/img]

2009 Ukraine National Mathematical Olympiad, 4

Tags: geometry , parallelogram , ratio , trigonometry

Let $ABCD$ be a parallelogram with $\angle BAC = 45^\circ,$ and $AC > BD .$ Let $w_1$ and $w_2$ be two circles with diameters $AC$ and $DC,$ respectively. The circle $w_1$ intersects $AB$ at $E$ and the circle $w_2$ intersects $AC$ at $O$ and $C$, and $AD$ at $F.$ Find the ratio of areas of triangles $AOE$ and $COF$ if $AO = a,$ and $FO = b .$

2024 All-Russian Olympiad, 6

Tags: parallelogram , geometry

Let $ABCD$ be a parallelogram. Let $M$ be the midpoint of the arc $AC$ containing $B$ of the circumcircle of $ABC$ . Let $E$ be a point on segment $AD$ and $F$ a point on segment $CD$ such that $ME=MD=MF$. Show that $BMEF$ is cyclic. [i]Proposed by A. Tereshin[/i]

1997 AMC 12/AHSME, 25

Tags: geometry , parallelogram

Let $ ABCD$ be a parallelogram and let $ \overrightarrow{AA^\prime}$, $ \overrightarrow{BB^\prime}$, $ \overrightarrow{CC^\prime}$, and $ \overrightarrow{DD^\prime}$ be parallel rays in space on the same side of the plane determined by $ ABCD$. If $ AA^\prime \equal{} 10$, $ BB^\prime \equal{} 8$, $ CC^\prime \equal{} 18$, $ DD^\prime \equal{} 22$, and $ M$ and $ N$ are the midpoints of $ \overline{A^{\prime}C^{\prime}}$ and $ \overline{B^{\prime}D^{\prime}}$, respectively, then $ MN \equal{}$ $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 3\qquad \textbf{(E)}\ 4$

1994 All-Russian Olympiad Regional Round, 9.6

Tags: parallelogram , geometry

Point $ P$ is taken inside a right angle $ KLM$. A circle $ S_1$ with center $ O_1$ is tangent to the rays $ LK,LP$ of angle $ KLP$ at $ A,D$ respectively. A circle $ S_2$ with center $ O_2$ is tangent to the rays of angle $ MLP$, touching $ LP$ at $ B$. Suppose $ A,B,O_1$ are collinear. Let $ O_2D,KL$ meet at $ C$. Prove that $ BC$ bisects angle $ ABD$.

2010 Indonesia TST, 2

Tags: length , inversion , geometry , homothety , parallelogram

Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $\Gamma_4$ be distinct circles such that $\Gamma_1$, $\Gamma_3$ are externally tangent at $P$, and $\Gamma_2$, $\Gamma_4$ are externally tangent at the same point $P$. Suppose that $\Gamma_1$ and $\Gamma_2$; $\Gamma_2$ and $\Gamma_3$; $\Gamma_3$ and $\Gamma_4$; $\Gamma_4$ and $\Gamma_1$ meet at $A$, $B$, $C$, $D$, respectively, and that all these points are different from $P$. Prove that \[ \frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}. \]

2010 Tuymaada Olympiad, 2

Tags: circumcircle , parallelogram , trigonometry , reflection , calculus , geometric transformation , geometry

Let $ABC$ be an acute triangle, $H$ its orthocentre, $D$ a point on the side $[BC]$, and $P$ a point such that $ADPH$ is a parallelogram. Show that $\angle BPC > \angle BAC$.

1999 Romania National Olympiad, 4

Tags: parallelogram , convexity , inequalities , algebra , geometry

a) Let $a,b\in R$, $a <b$. Prove that $x \in (a,b)$ if and only if there exists $\lambda \in (0,1)$ such that $x=\lambda a +(1-\lambda)b$. b) If the function $f: R \to R$ has the property: $$f (\lambda x+(1-\lambda) y) < \lambda f(x) + (1-\lambda)f(y), \forall x,y \in R, x\ne y, \forall \lambda \in (0,1), $$ prove that one cannot find four points on the function’s graph that are the vertices of a parallelogram

1966 IMO Shortlist, 17

Tags: locus problems , locus , parallelogram , geometry , vector

Let $ABCD$ and $A^{\prime }B^{\prime}C^{\prime }D^{\prime }$ be two arbitrary parallelograms in the space, and let $M,$ $N,$ $P,$ $Q$ be points dividing the segments $AA^{\prime },$ $BB^{\prime },$ $CC^{\prime },$ $DD^{\prime }$ in equal ratios. [b]a.)[/b] Prove that the quadrilateral $MNPQ$ is a parallelogram. [b]b.)[/b] What is the locus of the center of the parallelogram $MNPQ,$ when the point $M$ moves on the segment $AA^{\prime }$ ? (Consecutive vertices of the parallelograms are labelled in alphabetical order.

2023 Malaysia IMONST 2, 6

Tags: parallelogram , geometry

Ivan has a parallelogram whose interior angles are $60^{\circ}, 120^{\circ}, 60^{\circ}, 120^{\circ}$ respectively, and all side lengths are integers. Is it possible that one of the diagonals has length $\sqrt{2024}$?

Russian TST 2022, P2

Tags: equal angles , parallelogram , geometry

In parallelogram $ABCD$ with acute angle $A$ a point $N$ is chosen on the segment $AD$, and a point $M$ on the segment $CN$ so that $AB = BM = CM$. Point $K$ is the reflection of $N$ in line $MD$. The line $MK$ meets the segment $AD$ at point $L$. Let $P$ be the common point of the circumcircles of $AMD$ and $CNK$ such that $A$ and $P$ share the same side of the line $MK$. Prove that $\angle CPM = \angle DPL$.

2013 Mexico National Olympiad, 2

Tags: parallelogram , geometry

Let $ABCD$ be a parallelogram with the angle at $A$ obtuse. Let $P$ be a point on segment $BD$. The circle with center $P$ passing through $A$ cuts line $AD$ at $A$ and $Y$ and cuts line $AB$ at $A$ and $X$. Line $AP$ intersects $BC$ at $Q$ and $CD$ at $R$. Prove $\angle XPY = \angle XQY + \angle XRY$.

Cono Sur Shortlist - geometry, 2003.G4

Tags: midpoint , geometry , parallelogram

In a triangle $ABC$ , let $P$ be a point on its circumscribed circle (on the arc $AC$ that does not contain $B$). Let $H,H_1,H_2$ and $H_3$ be the orthocenters of triangles $ABC, BCP, ACP$ and $ABP$, respectively. Let $L = PB \cap AC$ and $J = HH_2 \cap H_1H_3$. If $M$ and $N$ are the midpoints of $JH$ and $LP$, respectively, prove that $MN$ and $JL$ intersect at their midpoint.

2006 Sharygin Geometry Olympiad, 8.3

Tags: parallelogram , equal segments , circles , geometry

A parallelogram $ABCD$ is given. Two circles with centers at the vertices $A$ and $C$ pass through $B$. The straight line $\ell$ that passes through $B$ and crosses the circles at second time at points $X, Y$ respectively. Prove that $DX = DY$.

1972 USAMO, 2

Tags: parallelogram , inequalities , 3d geometry , triangle inequality , tetrahedron , geometry

A given tetrahedron $ ABCD$ is isoceles, that is, $ AB\equal{}CD$, $ AC\equal{}BD$, $ AD\equal{}BC$. Show that the faces of the tetrahedron are acute-angled triangles.

2013 NIMO Problems, 3

Tags: geometric transformation , parallelogram , reflection , geometry

Let $ABC$ be a triangle. Prove that there exists a unique point $P$ for which one can find points $D$, $E$ and $F$ such that the quadrilaterals $APBF$, $BPCD$, $CPAE$, $EPFA$, $FPDB$, and $DPEC$ are all parallelograms. [i]Proposed by Lewis Chen[/i]

2012 CentroAmerican, 3

Tags: geometry , parallelogram , incenter

Let $ABC$ be a triangle with $AB < BC$, and let $E$ and $F$ be points in $AC$ and $AB$ such that $BF = BC = CE$, both on the same halfplane as $A$ with respect to $BC$. Let $G$ be the intersection of $BE$ and $CF$. Let $H$ be a point in the parallel through $G$ to $AC$ such that $HG = AF$ (with $H$ and $C$ in opposite halfplanes with respect to $BG$). Show that $\angle EHG = \frac{\angle BAC}{2}$.

1993 Moldova Team Selection Test, 8

Tags: geometry , circumcircle , parallelogram

Inside the parallelogram $ABCD$ points $M, N, K$ and $L{}$ are on sides $AB, BC, CD{}$ and $DA$, respectively. Let $O_1, O_2, O_3$ and $O_4$ be the circumcenters of triangles repesctively $MBN, NCK, KDL$ and $LAM{}$. Prove that the quadrilateral $O_1O_2O_3O_4$ is a parallelogram.

1998 All-Russian Olympiad, 2

Tags: parallelogram , geometry

A convex polygon is partitioned into parallelograms. A vertex of the polygon is called [i]good[/i] if it belongs to exactly one parallelogram. Prove that there are more than two good vertices.

2011 Romania Team Selection Test, 1

Tags: geometry , parallelogram , reflection , geometric transformation

Let $ABCD$ be a cyclic quadrilateral. The lines $BC$ and $AD$ meet at a point $P$. Let $Q$ be the point on the line $BP$, different from $B$, such that $PQ=BP$. Consider the parallelograms $CAQR$ and $DBCS$. Prove that the points $C,Q,R,S$ lie on a circle.

1988 AMC 12/AHSME, 6

Tags: rectangle , rhombus , trapezoid , geometry , parallelogram

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} $

2011 IMO Shortlist, 3

Tags: circumcircle , parallelogram , power of a point , geometry , perpendicular bisector

Let $ABCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB,BC$ and $CD$. Let $\omega_F$ be the circle through the feet of the perpendiculars from $F$ to the lines $CD,DA$ and $AB$. Prove that the midpoint of the segment $EF$ lies on the line through the two intersections of $\omega_E$ and $\omega_F$. [i]Proposed by Carlos Yuzo Shine, Brazil[/i]

2003 Estonia National Olympiad, 3

Tags: equal segments , altitude , parallelogram , geometry

In the acute-angled triangle $ABC$ all angles are greater than $45^o$. Let $AM$ and $BN$ be the heights of this triangle and let $X$ and $Y$ be the points on $MA$ and $NB$, respecively, such that $|MX| =|MB|$ and $|NY| =|NA|$. Prove that $MN$ and $XY$ are parallel.

Kyiv City MO Seniors 2003+ geometry, 2011.11.4

Tags: equal circles , circumcircle , parallelogram , geometry , cyclic quadrilateral

On the diagonals $AC$ and $BD$ of the inscribed quadrilateral A$BCD$, the points $X$ and $Y$ are marked, respectively, so that the quadrilateral $ABXY$ is a parallelogram. Prove that the circumscribed circles of triangles $BXD$ and $CYA$ have equal radii. (Vyacheslav Yasinsky)