Olympiad problems

2014 Contests, 3

The triangle $ABC$ is inscribed in a circle $w_1$. Inscribed in a triangle circle touchs the sides $BC$ in a point $N$. $w_2$ — the circle inscribed in a segment $BAC$ circle of $w_1$, and passing through a point $N$. Let points $O$ and $J$ — the centers of circles $w_2$ and an extra inscribed circle (touching side $BC$) respectively. Prove, that lines $AO$ and $JN$ are parallel.

2007 Indonesia TST, 1

Tags: trigonometry , geometry , circumcircle , geometry proposed

Let $ P$ be a point in triangle $ ABC$, and define $ \alpha,\beta,\gamma$ as follows: \[ \alpha\equal{}\angle BPC\minus{}\angle BAC, \quad \beta\equal{}\angle CPA\minus{}\angle \angle CBA, \quad \gamma\equal{}\angle APB\minus{}\angle ACB.\] Prove that \[ PA\dfrac{\sin \angle BAC}{\sin \alpha}\equal{}PB\dfrac{\sin \angle CBA}{\sin \beta}\equal{}PC\dfrac{\sin \angle ACB}{\sin \gamma}.\]

2009 Vietnam Team Selection Test, 1

Tags: geometry , geometric transformation , reflection , circumcircle , Euler , homothety , geometry proposed

Let an acute triangle $ ABC$ with curcumcircle $ (O)$. Call $ A_1,B_1,C_1$ are foots of perpendicular line from $ A,B,C$ to opposite side. $ A_2,B_2,C_2$ are reflect points of $ A_1,B_1,C_1$ over midpoints of $ BC,CA,AB$ respectively. Circle $ (AB_2C_2),(BC_2A_2),(CA_2B_2)$ cut $ (O)$ at $ A_3,B_3,C_3$ respectively. Prove that: $ A_1A_3,B_1B_3,C_1C_3$ are concurent.

2008 Tuymaada Olympiad, 4

Tags: geometry , geometric transformation , reflection , incenter , circumcircle , geometry proposed

Point $ I_1$ is the reflection of incentre $ I$ of triangle $ ABC$ across the side $ BC$. The circumcircle of $ BCI_1$ intersects the line $ II_1$ again at point $ P$. It is known that $ P$ lies outside the incircle of the triangle $ ABC$. Two tangents drawn from $ P$ to the latter circle touch it at points $ X$ and $ Y$. Prove that the line $ XY$ contains a medial line of the triangle $ ABC$. [i]Author: L. Emelyanov[/i]

2013 Greece Team Selection Test, 2

Tags: geometry , circumcircle , geometry proposed

Let $ABC$ be a non-isosceles,aqute triangle with $AB<AC$ inscribed in circle $c(O,R)$.The circle $c_{1}(B,AB)$ crosses $AC$ at $K$ and $c$ at $E$. $KE$ crosses $c$ at $F$ and $BO$ crosses $KE$ at $L$ and $AC$ at $M$ while $AE$ crosses $BF$ at $D$.Prove that: i)$D,L,M,F$ are concyclic. ii)$B,D,K,M,E$ are concyclic.

2013 Waseda University Entrance Examination, 5

Tags: geometry , 3D geometry , tetrahedron , trigonometry , area of a triangle , geometry proposed

Given a plane $P$ in space. For a figure $A$, call orthogonal projection the whole of points of intersection between the perpendicular drawn from each point in $A$ and $P$. Answer the following questions. (1) Let a plane $Q$ intersects with the plane $P$ by angle $\theta\ \left(0<\theta <\frac{\pi}{2}\right)$ between the planes, that is to say,　the angles between two lines, is $\theta$, which can be generated by cuttng $P,\ Q$ by a plane which is perpendicular to the line of intersection of $P$ and $Q$.　Find the maximum and minimum length of the orthogonal projection of the line segment in length 1 on $Q$ on to $P$.. (2) Consider $Q$ in (1). Find the area of the orthogonal projection of a equilateral triangle on $Q$ with side length 1 onto $P$. (3) What's the shape of the orthogonal projection $T'$ of a regular tetrahedron $T$ with side length 1 on to $P'$, then find the max area of $T'$.

1976 Canada National Olympiad, 6

Tags: geometry proposed , geometry

If $ A,B,C,D$ are four points in space, such that \[ \angle ABC\equal{}\angle BCD\equal{}\angle CDA\equal{}\angle DAB\equal{}\pi/2, \] prove that $ A,B,C,D$ lie in a plane.

2023 German National Olympiad, 2

Tags: geometry , geometry proposed , area of a triangle , areas , area , geometric inequality

In a triangle, the edges are extended past both vertices by the length of the edge opposite to the respective vertex. Show that the area of the resulting hexagon is at least $13$ times the area of the original triangle.

Brazil L2 Finals (OBM) - geometry, 1998.2

Tags: geometry proposed , geometry , Mass Points , center of mass , similar triangles

Let $ABC$ be a triangle. $D$ is the midpoint of $AB$, $E$ is a point on the side $BC$ such that $BE = 2 EC$ and $\angle ADC = \angle BAE$. Find $\angle BAC$.

1987 Romania Team Selection Test, 6

Tags: geometry , geometric transformation , rotation , geometry proposed

The plane is covered with network of regular congruent disjoint hexagons. Prove that there cannot exist a square which has its four vertices in the vertices of the hexagons. [i]Gabriel Nagy[/i]

2007 China Western Mathematical Olympiad, 4

Tags: inequalities , vector , geometry proposed , geometry

Let $ O$ be an interior point of the triangle $ ABC$. Prove that there exist positive integers $ p,q$ and $ r$ such that \[ |p\cdot\overrightarrow{OA} \plus{} q\cdot\overrightarrow{OB} \plus{} r\cdot\overrightarrow{OC}|<\frac{1}{2007}\]

2006 Germany Team Selection Test, 3

Tags: geometry proposed , geometry

Does there exist a set $ M$ of points in space such that every plane intersects $ M$ at a finite but nonzero number of points?

1990 IMO Longlists, 71

Tags: geometry proposed , geometry

Given a point $P = (p_1, p_2, \ldots, p_n)$ in $n$-dimensional space . Find point $X = (x_1, x_2, \ldots, x_n)$, such that $x_1 \leq x_2 \leq\cdots \leq x_n$ and $\sqrt{(x_1-p_1)^2 + (x_2-p_2)^2+\cdots+(x_n-p_n)^2}$ is minimal.

2012 Iran MO (3rd Round), 4

Tags: geometry , circumcircle , MIT , college , angle bisector , geometry proposed , Iran

The incircle of triangle $ABC$ for which $AB\neq AC$, is tangent to sides $BC,CA$ and $AB$ in points $D,E$ and $F$ respectively. Perpendicular from $D$ to $EF$ intersects side $AB$ at $X$, and the second intersection point of circumcircles of triangles $AEF$ and $ABC$ is $T$. Prove that $TX\perp TF$. [i]Proposed By Pedram Safaei[/i]

2008 Sharygin Geometry Olympiad, 17

Tags: geometry , incenter , parallelogram , geometry proposed

(A.Myakishev, 9--11) Given triangle $ ABC$ and a ruler with two marked intervals equal to $ AC$ and $ BC$. By this ruler only, find the incenter of the triangle formed by medial lines of triangle $ ABC$.

2006 Kyiv Mathematical Festival, 3

Tags: geometry , circumcircle , incenter , trigonometry , trig identities , Law of Sines , geometry proposed

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $O$ be the circumcenter and $H$ be the intersection point of the altitudes of acute triangle $ABC.$ The straight lines $BH$ and $CH$ intersect the segments $CO$ and $BO$ at points $D$ and $E$ respectively. Prove that if triangles $ODH$ and $OEH$ are isosceles then triangle $ABC$ is isosceles too.

2024 All-Russian Olympiad, 1

Tags: geometry , 3D geometry , tetrahedron , geometry proposed

We are given an infinite cylinder in space (i.e. the locus of points of a given distance $R>0$ from a given straight line). Can six straight lines containing the edges of a tetrahedron all have exactly one common point with this cylinder? [i]Proposed by A. Kuznetsov[/i]

2013 Saint Petersburg Mathematical Olympiad, 6

Tags: geometric transformation , geometry , circumcircle , geometry proposed

Let $(I_b)$, $(I_c)$ are excircles of a triangle $ABC$. Given a circle $ \omega $ passes through $A$ and externally tangents to the circles $(I_b)$ and $(I_c)$ such that it intersects with $BC$ at points $M$, $N$. Prove that $ \angle BAM=\angle CAN $. A. Smirnov

JBMO Geometry Collection, 2010

Tags: geometry , circumcircle , trigonometry , perpendicular bisector , angle bisector , geometry proposed

Let $AL$ and $BK$ be angle bisectors in the non-isosceles triangle $ABC$ ($L$ lies on the side $BC$, $K$ lies on the side $AC$). The perpendicular bisector of $BK$ intersects the line $AL$ at point $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK$. Prove that $LN = NA$.

1974 IMO Longlists, 47

Tags: ratio , geometry proposed , geometry

Given two points $A,B$ outside of a given plane $P,$ find the positions of points $M$ in the plane $P$ for which the ratio $\frac{MA}{MB}$ takes a minimum or maximum.

2002 China Western Mathematical Olympiad, 2

Tags: geometry , parallelogram , circumcircle , geometry proposed

Let $ O$ be the circumcenter of acute triangle $ ABC$. Point $ P$ is in the interior of triangle $ AOB$. Let $ D,E,F$ be the projections of $ P$ on the sides $ BC,CA,AB$, respectively. Prove that the parallelogram consisting of $ FE$ and $ FD$ as its adjacent sides lies inside triangle $ ABC$.

2012 Romanian Masters In Mathematics, 2

Tags: geometry , circumcircle , trigonometry , complex numbers , geometry proposed

Given a non-isosceles triangle $ABC$, let $D,E$, and $F$ denote the midpoints of the sides $BC,CA$, and $AB$ respectively. The circle $BCF$ and the line $BE$ meet again at $P$, and the circle $ABE$ and the line $AD$ meet again at $Q$. Finally, the lines $DP$ and $FQ$ meet at $R$. Prove that the centroid $G$ of the triangle $ABC$ lies on the circle $PQR$. [i](United Kingdom) David Monk[/i]

2019 IberoAmerican, 4

Tags: geometry , trapezoid , geometry solved , Iberoamerican , projective geometry , Miquel point , geometry proposed

Let $ABCD$ be a trapezoid with $AB\parallel CD$ and inscribed in a circumference $\Gamma$. Let $P$ and $Q$ be two points on segment $AB$ ($A$, $P$, $Q$, $B$ appear in that order and are distinct) such that $AP=QB$. Let $E$ and $F$ be the second intersection points of lines $CP$ and $CQ$ with $\Gamma$, respectively. Lines $AB$ and $EF$ intersect at $G$. Prove that line $DG$ is tangent to $\Gamma$.

2014 Contests, 3

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

Let $\Gamma_1$ be a circle and $P$ a point outside of $\Gamma_1$. The tangents from $P$ to $\Gamma_1$ touch the circle at $A$ and $B$. Let $M$ be the midpoint of $PA$ and $\Gamma_2$ the circle through $P$, $A$ and $B$. Line $BM$ cuts $\Gamma_2$ at $C$, line $CA$ cuts $\Gamma_1$ at $D$, segment $DB$ cuts $\Gamma_2$ at $E$ and line $PE$ cuts $\Gamma_1$ at $F$, with $E$ in segment $PF$. Prove lines $AF$, $BP$, and $CE$ are concurrent.

2014 CentroAmerican, 2

Tags: geometry , trapezoid , geometry proposed

Let $ABCD$ be a trapezoid with bases $AB$ and $CD$, inscribed in a circle of center $O$. Let $P$ be the intersection of the lines $BC$ and $AD$. A circle through $O$ and $P$ intersects the segments $BC$ and $AD$ at interior points $F$ and $G$, respectively. Show that $BF=DG$.