Olympiad problems

2006 Estonia Math Open Junior Contests, 3

Let ABCD be a parallelogram, M the midpoint of AB and N the intersection of CD and the angle bisector of ABC. Prove that CM and BN are perpendicular iff AN is the angle bisector of DAB.

2000 Bundeswettbewerb Mathematik, 3

Tags: geometry , 3D geometry , sphere , tetrahedron , geometry unsolved

For each vertex of a given tetrahedron, a sphere passing through that vertex and the midpoints of the edges outgoing from this vertex is constructed. Prove that these four spheres pass through a single point.

1986 IMO Longlists, 54

Tags: geometry unsolved , geometry

Find the least integer $n$ with the following property: For any set $V$ of $8$ points in the plane, no three lying on a line, and for any set $E$ of n line segments with endpoints in $V$ , one can find a straight line intersecting at least $4$ segments in $E$ in interior points.

2004 China Team Selection Test, 1

Tags: geometry unsolved , geometry

Using $ AB$ and $ AC$ as diameters, two semi-circles are constructed respectively outside the acute triangle $ ABC$. $ AH \perp BC$ at $ H$, $ D$ is any point on side $ BC$ ($ D$ is not coinside with $ B$ or $ C$ ), through $ D$, construct $ DE \parallel AC$ and $ DF \parallel AB$ with $ E$ and $ F$ on the two semi-circles respectively. Show that $ D$, $ E$, $ F$ and $ H$ are concyclic.

1972 IMO Longlists, 3

Tags: geometry unsolved , geometry

On a line a set of segments is given of total length less than $n$. Prove that every set of $n$ points of the line can be translated in some direction along the line for a distance smaller than $\frac{n}{2}$ so that none of the points remain on the segments.

1989 India National Olympiad, 6

Tags: geometry , incenter , angle bisector , perpendicular bisector , geometry unsolved

Triangle $ ABC$ has incentre $ I$ and the incircle touches $ BC, CA$ at $ D, E$ respectively. Let $ BI$ meet $ DE$ at $ G$. Show that $ AG$ is perpendicular to $ BG$.

2008 Sharygin Geometry Olympiad, 7

Tags: geometry , power of a point , radical axis , geometry unsolved

(F.Nilov) Two arcs with equal angular measure are constructed on the medians $ AA'$ and $ BB'$ of triangle $ ABC$ towards vertex $ C$. Prove that the common chord of the respective circles passes through $ C$.

2008 Mediterranean Mathematics Olympiad, 1

Tags: geometry unsolved , geometry

Let $ABCDEF$ be a convex hexagon such that all of its vertices are on a circle. Prove that $AD$, $BE$ and $CF$ are concurrent if and only if $\frac {AB}{BC}\cdot\frac {CD}{DE}\cdot\frac {EF}{FA}= 1$.

2010 Singapore Senior Math Olympiad, 1

Tags: geometry unsolved , geometry

In the $\triangle ABC$ with $AC>BC$ and $\angle B<90^{\circ}$, $D$ is the foot of the perpendicular from $A$ onto $BC$ and $E$ is the foot of perpendicular from $D$ onto $AC$. Let $F$ be the point on the segment $DE$ such that $EF \cdot DC=BD \cdot DE$. Prove that $AF$ is perpendicular to $BF$.

2008 Sharygin Geometry Olympiad, 5

Tags: geometry unsolved , geometry

(A.Zaslavsky) Given two triangles $ ABC$, $ A'B'C'$. Denote by $ \alpha$ the angle between the altitude and the median from vertex $ A$ of triangle $ ABC$. Angles $ \beta$, $ \gamma$, $ \alpha'$, $ \beta'$, $ \gamma'$ are defined similarly. It is known that $ \alpha \equal{} \alpha'$, $ \beta \equal{} \beta'$, $ \gamma \equal{} \gamma'$. Can we conclude that the triangles are similar?

2013 ELMO Shortlist, 6

Tags: trigonometry , projective geometry , geometry unsolved , geometry

Let $ABCDEF$ be a non-degenerate cyclic hexagon with no two opposite sides parallel, and define $X=AB\cap DE$, $Y=BC\cap EF$, and $Z=CD\cap FA$. Prove that \[\frac{XY}{XZ}=\frac{BE}{AD}\frac{\sin |\angle{B}-\angle{E}|}{\sin |\angle{A}-\angle{D}|}.\][i]Proposed by Victor Wang[/i]

2006 Silk Road, 4

Tags: induction , geometry unsolved , geometry

A family $L$ of 2006 lines on the plane is given in such a way that it doesn't contain parallel lines and it doesn't contain three lines with a common point.We say that the line $l_1\in L$ is [i]bounding[/i] the line $l_2\in L$,if all intersection points of the line $l_2$ with other lines from $L$ lie on the one side of the line $l_1$. Prove that in the family $L$ there are two lines $l$ and $l'$ such that the following 2 conditions are satisfied simultaneously: [b]1)[/b] The line $l$ is bounding the line $l'$; [b]2)[/b] the line $l'$ is not bounding the line $l$.

2009 India National Olympiad, 1

Tags: geometry , rectangle , parallelogram , trapezoid , trigonometry , perpendicular bisector , geometry unsolved

Let $ ABC$ be a tringle and let $ P$ be an interior point such that $ \angle BPC \equal{} 90 ,\angle BAP \equal{} \angle BCP$.Let $ M,N$ be the mid points of $ AC,BC$ respectively.Suppose $ BP \equal{} 2PM$.Prove that $ A,P,N$ are collinear.

2004 CHKMO, 3

Tags: geometry , parallelogram , analytic geometry , cyclic quadrilateral , geometry unsolved

Let $K, L, M, N$ be the midpoints of sides $AB, BC, CD, DA$ of a cyclic quadrilateral $ABCD$. Prove that the orthocentres of triangles $ANK, BKL, CLM, DMN$ are the vertices of a parallelogram.

2004 China Team Selection Test, 2

Tags: geometry , incenter , perimeter , geometric transformation , reflection , geometry unsolved

Two equal-radii circles with centres $ O_1$ and $ O_2$ intersect each other at $ P$ and $ Q$, $ O$ is the midpoint of the common chord $ PQ$. Two lines $ AB$ and $ CD$ are drawn through $ P$ ( $ AB$ and $ CD$ are not coincide with $ PQ$ ) such that $ A$ and $ C$ lie on circle $ O_1$ and $ B$ and $ D$ lie on circle $ O_2$. $ M$ and $ N$ are the mipoints of segments $ AD$ and $ BC$ respectively. Knowing that $ O_1$ and $ O_2$ are not in the common part of the two circles, and $ M$, $ N$ are not coincide with $ O$. Prove that $ M$, $ N$, $ O$ are collinear.

2012 CentroAmerican, 3

Tags: geometry , incenter , parallelogram , geometry unsolved

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 $\angle EHG = \frac{\angle BAC}{2}$.

2010 Sharygin Geometry Olympiad, 20

Tags: geometry , circumcircle , geometry unsolved

The incircle of an acute-angled triangle $ABC$ touches $AB, BC, CA$ at points $C_1, A_1, B_1$ respectively. Points $A_2, B_2$ are the midpoints of the segments $B_1C_1, A_1C_1$ respectively. Let $P$ be a common point of the incircle and the line $CO$, where $O$ is the circumcenter of triangle $ABC.$ Let also $A'$ and $B'$ be the second common points of $PA_2$ and $PB_2$ with the incircle. Prove that a common point of $AA'$ and $BB'$ lies on the altitude of the triangle dropped from the vertex $C.$

2019 Greece Team Selection Test, 2

Tags: geometry unsolved , geometry , circumcircle

Let a triangle $ABC$ inscribed in a circle $\Gamma$ with center $O$. Let $I$ the incenter of triangle $ABC$ and $D, E, F$ the contact points of the incircle with sides $BC, AC, AB$ of triangle $ABC$ respectively . Let also $S$ the foot of the perpendicular line from $D$ to the line $EF$.Prove that line $SI$ passes from the antidiametric point $N$ of $A$ in the circle $\Gamma$.( $AN$ is a diametre of the circle $\Gamma$).

2009 Iran MO (3rd Round), 5

Tags: geometry unsolved , geometry

5-Two circles $ S_1$ and $ S_2$ with equal radius and intersecting at two points are given in the plane.A line $ l$ intersects $ S_1$ at $ B,D$ and $ S_2$ at $ A,C$(the order of the points on the line are as follows:$ A,B,C,D$).Two circles $ W_1$ and $ W_2$ are drawn such that both of them are tangent externally at $ S_1$ and internally at $ S_2$ and also tangent to $ l$ at both sides.Suppose $ W_1$ and $ W_2$ are tangent.Then PROVE $ AB \equal{} CD$.

1995 Turkey Team Selection Test, 3

Tags: geometry , circumcircle , geometry unsolved

Let $D$ be a point on the small arc $AC$ of the circumcircle of an equilateral triangle $ABC$, different from $A$ and $C$. Let $E$ and $F$ be the projections of $D$ onto $BC$ and $AC$ respectively. Find the locus of the intersection point of $EF$ and $OD$, where $O$ is the center of $ABC$.

2003 China Western Mathematical Olympiad, 2

Tags: geometry , rectangle , parallelogram , geometry unsolved

A circle can be inscribed in the convex quadrilateral $ ABCD$. The incircle touches the sides $ AB, BC, CD, DA$ at $ A_1, B_1, C_1, D_1$ respectively. The points $ E, F, G, H$ are the midpoints of $ A_1B_1, B_1C_1, C_1D_1, D_1A_1$ respectively. Prove that the quadrilateral $ EFGH$ is a rectangle if and only if $ A, B, C, D$ are concyclic.

2001 Tournament Of Towns, 3

Tags: geometry , parallelogram , geometry unsolved

Let $AH_A$, $BH_B$ and $CH_C$ be the altitudes of triangle $\triangle ABC$. Prove that the triangle whose vertices are the intersection points of the altitudes of $\triangle AH_BH_C$, $\triangle BH_AH_C$ and $\triangle CH_AH_B$ is congruent to $\triangle H_AH_BH_C$.

1987 China Team Selection Test, 2

Tags: analytic geometry , geometry , induction , geometry unsolved , Polygons , coordinates , China

A closed recticular polygon with 100 sides (may be concave) is given such that it's vertices have integer coordinates, it's sides are parallel to the axis and all it's sides have odd length. Prove that it's area is odd.

1990 China Team Selection Test, 2

Tags: geometry unsolved , geometry

Finitely many polygons are placed in the plane. If for any two polygons of them, there exists a line through origin $O$ that cuts them both, then these polygons are called "properly placed". Find the least $m \in \mathbb{N}$, such that for any group of properly placed polygons, $m$ lines can drawn through $O$ and every polygon is cut by at least one of these $m$ lines.

2006 Hungary-Israel Binational, 1

Tags: quadratics , geometry , power of a point , geometry unsolved

A point $ P$ inside a circle is such that there are three chords of the same length passing through $ P$. Prove that $ P$ is the center of the circle.