Olympiad problems

2012 Brazil Team Selection Test, 4

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

Let $ ABC $ be an acute triangle. Denote by $ D $ the foot of the perpendicular line drawn from the point $ A $ to the side $ BC $, by $M$ the midpoint of $ BC $, and by $ H $ the orthocenter of $ ABC $. Let $ E $ be the point of intersection of the circumcircle $ \Gamma $ of the triangle $ ABC $ and the half line $ MH $, and $ F $ be the point of intersection (other than $E$) of the line $ ED $ and the circle $ \Gamma $. Prove that $ \tfrac{BF}{CF} = \tfrac{AB}{AC} $ must hold. (Here we denote $XY$ the length of the line segment $XY$.)

2006 Iran MO (3rd Round), 2

Tags: circumcircle , geometry , parallelogram , geometric transformation , rectangle

$ABC$ is a triangle and $R,Q,P$ are midpoints of $AB,AC,BC$. Line $AP$ intersects $RQ$ in $E$ and circumcircle of $ABC$ in $F$. $T,S$ are on $RP,PQ$ such that $ES\perp PQ,ET\perp RP$. $F'$ is on circumcircle of $ABC$ that $FF'$ is diameter. The point of intersection of $AF'$ and $BC$ is $E'$. $S',T'$ are on $AB,AC$ that $E'S'\perp AB,E'T'\perp AC$. Prove that $TS$ and $T'S'$ are perpendicular.

2009 AMC 12/AHSME, 22

Tags: parallelogram , geometry , analytic geometry , vector

Parallelogram $ ABCD$ has area $ 1,\!000,\!000$. Vertex $ A$ is at $ (0,0)$ and all other vertices are in the first quadrant. Vertices $ B$ and $ D$ are lattice points on the lines $ y\equal{}x$ and $ y\equal{}kx$ for some integer $ k>1$, respectively. How many such parallelograms are there? $ \textbf{(A)}\ 49\qquad \textbf{(B)}\ 720\qquad \textbf{(C)}\ 784\qquad \textbf{(D)}\ 2009\qquad \textbf{(E)}\ 2048$

2018 Middle European Mathematical Olympiad, 5

Tags: parallelogram , geometry , ohm

Let $ABC$ be an acute-angled triangle with $AB<AC,$ and let $D$ be the foot of its altitude from$A,$ points $B'$ and $C'$ lie on the rays $AB$ and $AC,$ respectively , so that points $B',$ $C'$ and $D$ are collinear and points $B,$ $C,$ $B'$ and $C'$ lie on one circle with center $O.$ Prove that if $M$ is the midpoint of $BC$ and $H$ is the orthocenter of $ABC,$ then $DHMO$ is a parallelogram.

2010 Canadian Mathematical Olympiad Qualification Repechage, 8

Tags: geometry , parallelogram

Consider three parallelograms $P_1,~P_2,~ P_3$. Parallelogram $P_3$ is inside parallelogram $P_2$, and the vertices of $P_3$ are on the edges of $P_2$. Parallelogram $P_2$ is inside parallelogram $P_1$, and the vertices of $P_2$ are on the edges of $P_1$. The sides of $P_3$ are parallel to the sides of $P_1$. Prove that one side of $P_3$ has length at least half the length of the parallel side of $P_1$.

Estonia Open Junior - geometry, 2014.2.5

Tags: point , parallelogram , concyclic , geometry

In the plane there are six different points $A, B, C, D, E, F$ such that $ABCD$ and $CDEF$ are parallelograms. What is the maximum number of those points that can be located on one circle?

2002 Baltic Way, 13

Tags: geometry , parallelogram , circumcircle

Let $ABC$ be an acute triangle with $\angle BAC>\angle BCA$, and let $D$ be a point on side $AC$ such that $|AB|=|BD|$. Furthermore, let $F$ be a point on the circumcircle of triangle $ABC$ such that line $FD$ is perpendicular to side $BC$ and points $F,B$ lie on different sides of line $AC$. Prove that line $FB$ is perpendicular to side $AC$ .

2012 France Team Selection Test, 2

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

Let $ABC$ be an acute-angled triangle with $AB\not= AC$. Let $\Gamma$ be the circumcircle, $H$ the orthocentre and $O$ the centre of $\Gamma$. $M$ is the midpoint of $BC$. The line $AM$ meets $\Gamma$ again at $N$ and the circle with diameter $AM$ crosses $\Gamma$ again at $P$. Prove that the lines $AP,BC,OH$ are concurrent if and only if $AH=HN$.

2006 CentroAmerican, 2

Tags: parallelogram , vector , trapezoid , geometric transformation , geometry

Let $\Gamma$ and $\Gamma'$ be two congruent circles centered at $O$ and $O'$, respectively, and let $A$ be one of their two points of intersection. $B$ is a point on $\Gamma$, $C$ is the second point of intersection of $AB$ and $\Gamma'$, and $D$ is a point on $\Gamma'$ such that $OBDO'$ is a parallelogram. Show that the length of $CD$ does not depend on the position of $B$.

1986 AIME Problems, 9

Tags: parallelogram , analytic geometry , symmetry , ratio , geometry

In $\triangle ABC$, $AB= 425$, $BC=450$, and $AC=510$. An interior point $P$ is then drawn, and segments are drawn through $P$ parallel to the sides of the triangle. If these three segments are of an equal length $d$, find $d$.

Ukraine Correspondence MO - geometry, 2017.11

Tags: parallelogram , geometry , tangent circle

Inside the parallelogram $ABCD$, choose a point $P$ such that $\angle APB+ \angle CPD= \angle BPC+ \angle APD$. Prove that there exists a circle tangent to each of the circles circumscribed around the triangles $APB$, $BPC$, $CPD$ and $APD$.

2005 JBMO Shortlist, 7

Tags: parallelogram , geometry

Let $ABCD$ be a parallelogram. $P \in (CD), Q \in (AB)$, $M= AP \cap DQ$, $N=BP \cap CQ$, $ K=MN \cap AD$, $L= MN \cap BC$. Prove that $BL=DK$.

Indonesia MO Shortlist - geometry, g3.3

Tags: logarithm , geometry , geometric transformation , dilation , trapezoid , trigonometry , parallelogram

Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.

2016 Macedonia JBMO TST, 2

Tags: geometry , parallelogram

Let $ABCD$ be a parallelogram and let $E$, $F$, $G$, and $H$ be the midpoints of sides $AB$, $BC$, $CD$, and $DA$, respectively. If $BH \cap AC = I$, $BD \cap EC = J$, $AC \cap DF = K$, and $AG \cap BD = L$, prove that the quadrilateral $IJKL$ is a parallelogram.

1997 Taiwan National Olympiad, 5

Tags: trigonometry , geometry , 3d geometry , tetrahedron , parallelogram

Let $ABCD$ is a tetrahedron. Show that a)If $AB=CD,AC=DB,AD=BC$ then triangles $ABC,ABD,ACD,BCD$ are acute. b)If the triangles $ABC,ABD,ACD,BCD$ have the same area , then $AB=CD,AC=DB,AD=BC$.

2020 Junior Macedonian National Olympiad, 4

Tags: geometry , nice , collinearity , circumcircle , parallelogram

Let $ABC$ be an isosceles triangle with base $AC$. Points $D$ and $E$ are chosen on the sides $AC$ and $BC$, respectively, such that $CD = DE$. Let $H, J,$ and $K$ be the midpoints of $DE, AE,$ and $BD$, respectively. The circumcircle of triangle $DHK$ intersects $AD$ at point $F$, whereas the circumcircle of triangle $HEJ$ intersects $BE$ at $G$. The line through $K$ parallel to $AC$ intersects $AB$ at $I$. Let $IH \cap GF =$ {$M$}. Prove that $J, M,$ and $K$ are collinear points.

1966 IMO Shortlist, 22

Tags: geometry , vector , parallelogram

Let $P$ and $P^{\prime }$ be two parallelograms with equal area, and let their sidelengths be $a,$ $b$ and $a^{\prime },$ $b^{\prime }.$ Assume that $a^{\prime }\leq a\leq b\leq b^{\prime },$ and moreover, it is possible to place the segment $b^{\prime }$ such that it completely lies in the interior of the parallelogram $P.$ Show that the parallelogram $P$ can be partitioned into four polygons such that these four polygons can be composed again to form the parallelogram $% P^{\prime }.$

1997 Romania National Olympiad, 4

Tags: geometry , parallelogram

The quadrilateral $ABCD$ has two parallel sides. Let $M$ and $N$ be the midpoints of $[DC]$ and $[BC]$, and $P$ the common point of the lines $AM$ and $DN$. If $\frac{PM}{AP}=\frac{1}{4}$, prove that $ABCD$ is a parallelogram.

2013 Bogdan Stan, 3

Tags: geometry , parallelogram

$ O $ is the center of a parallelogram $ ABCD. $ Let $ G $ on the segment $ OB $ (excluding its endpoints), $ N $ on the line $ DC $ and $ M $ on the segment $ AD $ (excluding its endpoints) such that $ CN>ND, AM=6MD $ and so that there exists a natural number $ n\ge 3 $ such that $ OB=nGO. $ Show that $ G,M,N $ are collinear if and only if $$ \left( \frac{CN}{ND} -6 \right) (n+1)=2. $$

2000 Tournament Of Towns, 2

Tags: congruent triangles , integer sides , geometry , parallelogram

Two parallel sides of a quadrilateral have integer lengths. Prove that this quadrilateral can be cut into congruent triangles. (A Shapovalov)

1972 IMO Longlists, 43

Tags: trapezoid , parallelogram , rectangle , geometry , pythagorean theorem , perpendicular bisector

A fixed point $A$ inside a circle is given. Consider all chords $XY$ of the circle such that $\angle XAY$ is a right angle, and for all such chords construct the point $M$ symmetric to $A$ with respect to $XY$ . Find the locus of points $M$.

2008 Hanoi Open Mathematics Competitions, 9

Tags: parallelogram , construction , equal angles , geometry

Consider a triangle $ABC$. For every point M $\in BC$ ,we define $N \in CA$ and $P \in AB$ such that $APMN$ is a parallelogram. Let $O$ be the intersection of $BN$ and $CP$. Find $M \in BC$ such that $\angle PMO=\angle OMN$

1983 Tournament Of Towns, (033) O2

Tags: rectangle , parallelogram , regular polygon , area , geometry

(a) A regular $4k$-gon is cut into parallelograms. Prove that among these there are at least $k$ rectangles. (b) Find the total area of the rectangles in (a) if the lengths of the sides of the $4k$-gon equal $a$. (VV Proizvolov, Moscow)

2024 Mozambique National Olympiad, P6

Tags: right triangle , geometry , area , parallelogram

Let $ABC$ be an isosceles right triangle with $\angle BCA=90^{\circ}, BC=AC=10$. Let $P$ be a point on $AB$ that is a distance $x$ from $A$, $Q$ be a point on $AC$ such that $PQ$ is parallel to $BC$. Let $R$ and $S$ be points on $BC$ such that $QR$ is parallel to $AB$ and $PS$ is parallel to $AC$. The union of the quadrilaterals $PBRQ$ and $PSCQ$ determine a shaded area $f(x)$. Evaluate $f(2)$

2023 New Zealand MO, 2

Tags: area , parallelogram , area of a triangle , geometry

Let $ABCD$ be a parallelogram, and let $P$ be a point on the side $AB$. Let the line through $P$ parallel to $BC$ intersect the diagonal $AC$ at point $Q$. Prove that $$|DAQ|^2 = |PAQ| \times |BCD| ,$$ where $|XY Z|$ denotes the area of triangle $XY Z$.