Olympiad problems

2000 Baltic Way, 1

Let $ K$ be a point inside the triangle $ ABC$. Let $ M$ and $ N$ be points such that $ M$ and $ K$ are on opposite sides of the line $ AB$, and $ N$ and $ K$ are on opposite sides of the line $ BC$. Assume that $ \angle MAB \equal{} \angle MBA \equal{} \angle NBC \equal{} \angle NCB \equal{} \angle KAC \equal{} \angle KCA$. Show that $ MBNK$ is a parallelogram.

2005 All-Russian Olympiad, 1

Tags: geometric transformation , circumcircle , perpendicular bisector , geometry , reflection , parallelogram , angle bisector

Given a parallelogram $ABCD$ with $AB<BC$, show that the circumcircles of the triangles $APQ$ share a second common point (apart from $A$) as $P,Q$ move on the sides $BC,CD$ respectively s.t. $CP=CQ$.

1977 IMO Longlists, 55

Tags: geometry , projective geometry , parallelogram

Through a point $O$ on the diagonal $BD$ of a parallelogram $ABCD$, segments $MN$ parallel to $AB$, and $PQ$ parallel to $AD$, are drawn, with $M$ on $AD$, and $Q$ on $AB$. Prove that diagonals $AO,BP,DN$ (extended if necessary) will be concurrent.

2013 ITAMO, 5

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

$ABC$ 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.

2019 Iran Team Selection Test, 3

Tags: geometry , parallelogram

Point $P$ lies inside of parallelogram $ABCD$. Perpendicular lines to $PA,PB,PC$ and $PD$ through $A,B,C$ and $D$ construct convex quadrilateral $XYZT$. Prove that $S_{XYZT}\geq 2S_{ABCD}$. [i]Proposed by Siamak Ahmadpour[/i]

2004 Iran MO (3rd Round), 16

Tags: cyclic quadrilateral , geometric transformation , parallelogram , rectangle , homothety , circumcircle , geometry

Let $ABC$ be a triangle . Let point $X$ be in the triangle and $AX$ intersects $BC$ in $Y$ . Draw the perpendiculars $YP,YQ,YR,YS$ to lines $CA,CX,BX,BA$ respectively. Find the necessary and sufficient condition for $X$ such that $PQRS$ be cyclic .

1988 Bundeswettbewerb Mathematik, 3

Tags: algebra , polynomial , trigonometry , geometry , parallelogram

Consider an octagon with equal angles and with rational sides. Prove that it has a center of symmetry.

2001 IMO Shortlist, 3

Tags: geometry , minimization , triangle , parallelogram

Let $ABC$ be a triangle with centroid $G$. Determine, with proof, the position of the point $P$ in the plane of $ABC$ such that $AP{\cdot}AG + BP{\cdot}BG + CP{\cdot}CG$ is a minimum, and express this minimum value in terms of the side lengths of $ABC$.

Champions Tournament Seniors - geometry, 2018.3

Tags: equal angles , equal segments , geometry , parallelogram

The vertex $F$ of the parallelogram $ACEF$ lies on the side $BC$ of parallelogram $ABCD$. It is known that $AC = AD$ and $AE = 2CD$. Prove that $\angle CDE = \angle BEF$.

1964 AMC 12/AHSME, 17

Tags: parallelogram , trapezoid , geometry

Given the distinct points $P(x_1, y_1)$, $Q(x_2, y_2)$ and $R(x_1+x_2, y_1+y_2)$. Line segments are drawn connecting these points to each other and to the origin $0$. Of the three possibilities: (1) parallelogram (2) straight line (3) trapezoid, figure $OPRQ$, depending upon the location of the points $P, Q,$ and $R$, can be: $ \textbf{(A)}\ \text{(1) only}\qquad\textbf{(B)}\ \text{(2) only}\qquad\textbf{(C)}\ \text{(3) only}\qquad\textbf{(D)}\ \text{(1) or (2) only}\qquad\textbf{(E)}\ \text{all three} $

2022 Cyprus TST, 3

Tags: parallelogram , circumcircle , geometry

Let $\triangle ABC$ be an acute-angled triangle with $AB<AC$ and let $(c)$ be its circumcircle with center $O$. Let $M$ be the midpoint of $BC$. The line $AM$ meets the circle $(c)$ again at the point $D$. The circumcircle $(c_1)$ of triangle $\triangle MDC$ intersects the line $AC$ at the points $C$ and $I$, and the circumcircle $(c_2)$ of $\triangle AMI$ intersects the line $AB$ at the points $A$ and $Z$. If $N$ is the foot of the perpendicular from $B$ on $AC$, and $P$ is the second point of intersection of $ZN$ with $(c_2)$, prove that the quadrilateral with vertices the points $N, P, I$ and $M$ is a parallelogram.

2010 Contests, 3

Tags: parallelogram , angle bisector , trigonometry , geometry

In plane,let a circle $(O)$ and two fixed points $B,C$ lies in $(O)$ such that $BC$ not is the diameter.Consider a point $A$ varies in $(O)$ such that $A\neq B,C$ and $AB\neq AC$.Call $D$ and $E$ respective is intersect of $BC$ and internal and external bisector of $\widehat{BAC}$,$I$ is midpoint of $DE$.The line that pass through orthocenter of $\triangle ABC$ and perpendicular with $AI$ intersects $AD,AE$ respective at $M,N$. 1/Prove that $MN$ pass through a fixed point 2/Determint the place of $A$ such that $S_{AMN}$ has maxium value

1972 IMO Longlists, 13

Tags: geometry , parallelogram , 3d geometry , sphere

Given a sphere $K$, determine the set of all points $A$ that are vertices of some parallelograms $ABCD$ that satisfy $AC \le BD$ and whose entire diagonal $BD$ is contained in $K$.

2022 IOQM India, 5

Tags: geometry , parallelogram

In parallelogram $ABCD$, the longer side is twice the shorter side. Let $XYZW$ be the quadrilateral formed by the internal bisectors of the angles of $ABCD$. If the area of $XYZW$ is $10$, find the area of $ABCD$

2005 Federal Competition For Advanced Students, Part 1, 4

Tags: reflection , homothety , geometric transformation , parallelogram , geometry

We're given two congruent, equilateral triangles $ABC$ and $PQR$ with parallel sides, but one has one vertex pointing up and the other one has the vertex pointing down. One is placed above the other so that the area of intersection is a hexagon $A_1A_2A_3A_4A_5A_6$ (labelled counterclockwise). Prove that $A_1A_4$, $A_2A_5$ and $A_3A_6$ are concurrent.

2000 Baltic Way, 4

Tags: geometry , parallelogram

Given a triangle $ ABC$ with $ \angle A \equal{} 120^{\circ}$. The points $ K$ and $ L$ lie on the sides $ AB$ and $ AC$, respectively. Let $ BKP$ and $ CLQ$ be equilateral triangles constructed outside the triangle $ ABC$. Prove that $ PQ \ge\frac{\sqrt 3}{2}\left(AB \plus{} AC\right)$.

2020 Vietnam National Olympiad, 4

Tags: geometry , reflection , parallelogram , geometric transformation

Let a non-isosceles acute triangle ABC with the circumscribed cycle (O) and the orthocenter H. D, E, F are the reflection of O in the lines BC, CA and AB. a) $H_a$ is the reflection of H in BC, A' is the reflection of A at O and $O_a$ is the center of (BOC). Prove that $H_aD$ and OA' intersect on (O). b) Let X is a point satisfy AXDA' is a parallelogram. Prove that (AHX), (ABF), (ACE) have a comom point different than A

1992 Baltic Way, 16

Tags: parallelogram , geometry

All faces of a convex polyhedron are parallelograms. Can the polyhedron have exactly 1992 faces?

2013 ELMO Shortlist, 7

Tags: ratio , trigonometry , complex , circumcircle , collinear , parallelogram , geometry

Let $ABC$ be a triangle inscribed in circle $\omega$, and let the medians from $B$ and $C$ intersect $\omega$ at $D$ and $E$ respectively. Let $O_1$ be the center of the circle through $D$ tangent to $AC$ at $C$, and let $O_2$ be the center of the circle through $E$ tangent to $AB$ at $B$. Prove that $O_1$, $O_2$, and the nine-point center of $ABC$ are collinear. [i]Proposed by Michael Kural[/i]

2014 Peru MO (ONEM), 4

Tags: parallelogram , geometry , circumcircle

Let $ABC$ be an acute triangle with circumcenter $O$, on the sides $BC, CA$ and $AB$ they take the points $D, E$ and $F$, respectively, in such a way that $BDEF$ is a parallelogram. Supposing that $DF^2 = AE\cdot EC <\frac{AC^2}{4}$ show that the circles circumscribed to the triangles $FBD$ and $AOC$ are tangent.

2022 Rioplatense Mathematical Olympiad, 4

Tags: geometry , parallelogram

Let $ABCD$ be a parallelogram and $M$ is the intersection of $AC$ and $BD$. The point $N$ is inside of the $\triangle AMB$ such that $\angle AND=\angle BNC$. Prove that $\angle MNC=\angle NDA$ and $\angle MND=\angle NCB$.

2001 Paraguay Mathematical Olympiad, 4

Tags: parallelogram , area , geometry

In a parallelogram $ABCD$ of surface area $60$ cm$^2$ , a line is drawn by $D$ that intersects $BC$ at $P$ and the extension of $AB$ at $Q$. If the area of the quadrilateral $ABPD$ is $46$ cm$^2$ , find the area of triangle $CPQ$.

2010 Turkey Team Selection Test, 2

Tags: geometry , circumcircle , analytic geometry , parallelogram

For an interior point $D$ of a triangle $ABC,$ let $\Gamma_D$ denote the circle passing through the points $A, \: E, \: D, \: F$ if these points are concyclic where $BD \cap AC=\{E\}$ and $CD \cap AB=\{F\}.$ Show that all circles $\Gamma_D$ pass through a second common point different from $A$ as $D$ varies.

Kyiv City MO 1984-93 - geometry, 1991.8.3

Tags: parallelogram , equilateral , geometry

On the sides of the parallelogram $ABCD$ outside it are constructed equilateral triangles $ABM$, $BCN$, $CDP$, $ADQ$. Prove that $MNPQ$ is a parallelogram.

2009 Dutch Mathematical Olympiad, 4

Tags: parallelogram , geometry , similar triangles

Let $ABC$ be an arbitrary triangle. On the perpendicular bisector of $AB$, there is a point $P$ inside of triangle $ABC$. On the sides $BC$ and $CA$, triangles $BQC$ and $CRA$ are placed externally. These triangles satisfy $\vartriangle BPA \sim \vartriangle BQC \sim \vartriangle CRA$. (So $Q$ and $A$ lie on opposite sides of $BC$, and $R$ and $B$ lie on opposite sides of $AC$.) Show that the points $P, Q, C$ and $R$ form a parallelogram.