Olympiad problems

1995 Taiwan National Olympiad, 6

Tags: analytic geometry , inequalities , geometry , vector , parallelogram , combinatorics

Let $a,b,c,d$ are integers such that $(a,b)=(c,d)=1$ and $ad-bc=k>0$. Prove that there are exactly $k$ pairs $(x_{1},x_{2})$ of rational numbers with $0\leq x_{1},x_{2}<1$ for which both $ax_{1}+bx_{2},cx_{1}+dx_{2}$ are integers.

2008 Sharygin Geometry Olympiad, 4

Tags: geometry , symmetry , angle bisector , parallelogram

(F.Nilov, A.Zaslavsky) Let $ CC_0$ be a median of triangle $ ABC$; the perpendicular bisectors to $ AC$ and $ BC$ intersect $ CC_0$ in points $ A'$, $ B'$; $ C_1$ is the meet of lines $ AA'$ and $ BB'$. Prove that $ \angle C_1CA \equal{} \angle C_0CB$.

2007 Vietnam Team Selection Test, 2

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

Let $ABC$ be an acute triangle with incricle $(I)$. $(K_{A})$ is the cricle such that $A\in (K_{A})$ and $AK_{A}\perp BC$ and it in-tangent for $(I)$ at $A_{1}$, similary we have $B_{1},C_{1}$. a) Prove that $AA_{1},BB_{1},CC_{1}$ are concurrent, called point-concurrent is $P$. b) Assume circles $(J_{A}),(J_{B}),(J_{C})$ are symmetry for excircles $(I_{A}),(I_{B}),(I_{C})$ across midpoints of $BC,CA,AB$ ,resp. Prove that $P_{P/(J_{A})}=P_{P/(J_{B})}=P_{P/(J_{C})}$. Note. If $(O;R)$ is a circle and $M$ is a point then $P_{M/(O)}=OM^{2}-R^{2}$.

2006 India IMO Training Camp, 2

Tags: exterior angle , spiral similarity , geometry , parallelogram , incenter , homothety

Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

PEN R Problems, 11

Tags: parallelogram , the geometry of numbers , geometry

Prove that if a lattice parallelogram contains at most three lattice points in addition to its vertices, then those are on one of the diagonals.

Kyiv City MO Juniors Round2 2010+ geometry, 2019.8.41

Tags: geometry , equal segments , parallelogram

Through the vertices $A, B$ of the parallelogram $ABCD$ passes a circle that intersects for the second time diagonals $BD$ and $AC$ at points $X$ and $Y$, respectively. The circumsccribed circle of $\vartriangle ADX$ intersects diagonal $AC$ for the second time at the point $Z$. Prove that $AY = CZ$.

1967 IMO Longlists, 8

Tags: trigonometric inequality , geometric inequality , geometry , trigonometry , parallelogram

The parallelogram $ABCD$ has $AB=a,AD=1,$ $\angle BAD=A$, and the triangle $ABD$ has all angles acute. Prove that circles radius $1$ and center $A,B,C,D$ cover the parallelogram if and only \[a\le\cos A+\sqrt3\sin A.\]

1998 All-Russian Olympiad Regional Round, 10.2

Tags: geometry , parallelogram , circumcircle

In an acute triangle $ABC$, a circle $S$ is drawn through the center $O$ of the circumcircle and the vertices $B$ and $C$. Let $OK$ be the diameter of the circle $S$, $D$ and $E$, be it's intersection points with the straight lines $AB$ and $AC$ respectively. Prove that $ADKE$ is a parallelogram.

2009 AMC 10, 23

Tags: trapezoid , ratio , geometry , trigonometry , parallelogram

Convex quadrilateral $ ABCD$ has $ AB\equal{}9$ and $ CD\equal{}12$. Diagonals $ AC$ and $ BD$ intersect at $ E$, $ AC\equal{}14$, and $ \triangle AED$ and $ \triangle BEC$ have equal areas. What is $ AE$? $ \textbf{(A)}\ \frac{9}{2}\qquad \textbf{(B)}\ \frac{50}{11}\qquad \textbf{(C)}\ \frac{21}{4}\qquad \textbf{(D)}\ \frac{17}{3}\qquad \textbf{(E)}\ 6$

1998 Federal Competition For Advanced Students, Part 2, 3

Tags: ratio , geometry , parallelogram

In a parallelogram $ABCD$ with the side ratio $AB : BC = 2 : \sqrt 3$ the normal through $D$ to $AC$ and the normal through $C$ to $AB$ intersects in the point $E$ on the line $AB$. What is the relationship between the lengths of the diagonals $AC$ and $BD$?

2007 IMO Shortlist, 4

Tags: geometry , circumcircle , homothety , parallelogram

Consider five points $ A$, $ B$, $ C$, $ D$ and $ E$ such that $ ABCD$ is a parallelogram and $ BCED$ is a cyclic quadrilateral. Let $ \ell$ be a line passing through $ A$. Suppose that $ \ell$ intersects the interior of the segment $ DC$ at $ F$ and intersects line $ BC$ at $ G$. Suppose also that $ EF \equal{} EG \equal{} EC$. Prove that $ \ell$ is the bisector of angle $ DAB$. [i]Author: Charles Leytem, Luxembourg[/i]

2008 Postal Coaching, 1

Tags: parallelogram , cyclic quadrilateral , concyclic , geometry

Let $ABCD$ be a quadrilateral that can be inscribed in a circle. Denote by $P$ the intersection point of lines $AD$ and $BC$, and by $Q$ the intersection point of lines $AB$ and $DC$. Let $E$ be the fourth vertex of the parallelogram $ABCE$, and $F$ the intersection of lines $CE$ is $PQ$. Prove that the points $D,E, F$, and $Q$ lie on the same circle.

2015 Bulgaria National Olympiad, 5

Tags: parallelogram , geometry , concurrency

In a triangle $\triangle ABC$ points $L, P$ and $Q$ lie on the segments $AB, AC$ and $BC$, respectively, and are such that $PCQL$ is a parallelogram. The circle with center the midpoint $M$ of the segment $AB$ and radius $CM$ and the circle of diameter $CL$ intersect for the second time at the point $T$. Prove that the lines $AQ, BP$ and $LT$ intersect in a point.

2023 Irish Math Olympiad, P1

Tags: parallelogram , congruent triangles , geometry

We are given a triangle $ABC$ such that $\angle BAC < 90^{\circ}$. The point $D$ is on the opposite side of the line $AB$ to $C$ such that $|AD| = |BD|$ and $\angle ADB = 90^{\circ}$. Similarly, the point $E$ is on the opposite side of $AC$ to $B$ such that $|AE| = |CE|$ and $\angle AEC = 90^{\circ}$. The point $X$ is such that $ADXE$ is a parallelogram. Prove that $|BX| = |CX|$.

2000 India Regional Mathematical Olympiad, 1

Tags: parallelogram , ibo 2k14 , analytic geometry

Let $AC$ be a line segment in the plane and $B$ a points between $A$ and $C$. Construct isosceles triangles $PAB$ and $QAC$ on one side of the segment $AC$ such that $\angle APB = \angle BQC = 120^{\circ}$ and an isosceles triangle $RAC$ on the other side of $AC$ such that $\angle ARC = 120^{\circ}.$ Show that $PQR$ is an equilateral triangle.

Estonia Open Senior - geometry, 2008.2.3

Tags: geometry , parallelogram , circles

Two circles are drawn inside a parallelogram $ABCD$ so that one circle is tangent to sides $AB$ and $AD$ and the other is tangent to sides $CB$ and $CD$. The circles touch each other externally at point $K$. Prove that $K$ lies on the diagonal $AC$.

2010 India IMO Training Camp, 10

Tags: trigonometry , parallelogram , complex number , geometry , analytic geometry , inequalities

Let $ABC$ be a triangle. Let $\Omega$ be the brocard point. Prove that $\left(\frac{A\Omega}{BC}\right)^2+\left(\frac{B\Omega}{AC}\right)^2+\left(\frac{C\Omega}{AB}\right)^2\ge 1$

JBMO Geometry Collection, 2014

Tags: geometry , parallelogram , analytic geometry , circumcircle

Consider an acute triangle $ABC$ of area $S$. Let $CD \perp AB$ ($D \in AB$), $DM \perp AC$ ($M \in AC$) and $DN \perp BC$ ($N \in BC$). Denote by $H_1$ and $H_2$ the orthocentres of the triangles $MNC$, respectively $MND$. Find the area of the quadrilateral $AH_1BH_2$ in terms of $S$.

2004 Bosnia and Herzegovina Team Selection Test, 6

Tags: parallelogram , geometry , concurrency

It is given triangle $ABC$ and parallelogram $ASCR$ with diagonal $AC$. Let line constructed through point $B$ parallel with $CS$ intersects line $AS$ and $CR$ in $M$ and $P$, respectively. Let line constructed through point $B$ parallel with $AS$ intersects line $AR$ and $CS$ in $N$ and $Q$, respectively. Prove that lines $RS$, $MN$ and $PQ$ are concurrent

2008 Baltic Way, 16

Tags: geometry , rhombus , parallelogram

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.

1993 Poland - First Round, 7

Tags: similar triangles , parallelogram , geometry

Given convex quadrilateral $ABCD$. We construct the similar triangles $APB, BQC, CRD, DSA$ outside $ABCD$ so that $\angle PAB = \angle QBC = \angle RCD = \angle SDA, \angle PBA = \angle QCB = \angle RDC = \angle SAD$. Prove that if $PQRS$ is a parallelogram, so is $ABCD$.

2003 Turkey Team Selection Test, 5

Tags: geometric transformation , geometry , reflection , parallelogram

Let $A$ be a point on a circle with center $O$ and $B$ be the midpoint of $[OA]$. Let $C$ and $D$ be points on the circle such that they lie on the same side of the line $OA$ and $\widehat{CBO} = \widehat{DBA}$. Show that the reflection of the midpoint of $[CD]$ over $B$ lies on the circle.

2014 Online Math Open Problems, 4

Tags: geometry , parallelogram

A crazy physicist has discovered a new particle called an emon. He starts with two emons in the plane, situated a distance $1$ from each other. He also has a crazy machine which can take any two emons and create a third one in the plane such that the three emons lie at the vertices of an equilateral triangle. After he has five total emons, let $P$ be the product of the $\binom 52 = 10$ distances between the $10$ pairs of emons. Find the greatest possible value of $P^2$. [i]Proposed by Yang Liu[/i]

1995 Argentina National Olympiad, 3

Tags: parallelogram , geometry

Let ABCD be a parallelogram, and P a point such that $2 PDA=ABP$ and $2 PAD=PCD$ Show that $AB=BP=CP$

1983 IMO Shortlist, 4

Tags: parallelogram , geometry , trigonometry , triangle

On the sides of the triangle $ABC$, three similar isosceles triangles $ABP \ (AP = PB)$, $AQC \ (AQ = QC)$, and $BRC \ (BR = RC)$ are constructed. The first two are constructed externally to the triangle $ABC$, but the third is placed in the same half-plane determined by the line $BC$ as the triangle $ABC$. Prove that $APRQ$ is a parallelogram.