Olympiad problems

2009 China Western Mathematical Olympiad, 3

Let $H$ be the orthocenter of acute triangle $ABC$ and $D$ the midpoint of $BC$. A line through $H$ intersects $AB,AC$ at $F,E$ respectively, such that $AE=AF$. The ray $DH$ intersects the circumcircle of $\triangle ABC$ at $P$. Prove that $P,A,E,F$ are concyclic.

2006 Moldova MO 11-12, 7

Tags: geometry proposed , geometry

Let $n\in\mathbb{N}^*$. $2n+3$ points on the plane are given so that no 3 lie on a line and no 4 lie on a circle. Is it possible to find 3 points so that the interior of the circle passing through them would contain exactly $n$ of the remaining points.

2001 Junior Balkan Team Selection Tests - Romania, 2

Tags: geometry , trapezoid , circumcircle , geometry proposed

Let $ABCDEF$ be a hexagon with $AB||DE,\ BC||EF,\ CD||FA$ and in which the diagonals $AD,BE$ and $CF$ are congruent. Prove that the hexagon can be inscribed in a circle.

2011 Baltic Way, 15

Tags: geometry proposed , geometry

Let $ABCD$ be a convex quadrilateral such that $\angle ADB=\angle BDC$. Suppose that a point $E$ on the side $AD$ satisfies the equality \[AE\cdot ED + BE^2=CD\cdot AE.\] Show that $\angle EBA=\angle DCB$.

1999 Italy TST, 2

Tags: geometry , inequalities , geometry proposed

Let $D$ and $E$ be points on sides $AB$ and $AC$ respectively of a triangle $ABC$ such that $DE$ is parallel to $BC$ and tangent to the incircle of $ABC$. Prove that \[DE\le\frac{1}{8}(AB+BC+CA) \]

1991 Turkey Team Selection Test, 1

Tags: ratio , geometry , geometry proposed

Let $C',B',A'$ be points respectively on sides $AB,AC,BC$ of $\triangle ABC$ satisfying $ \tfrac{AB'}{B'C}= \tfrac{BC'}{C'A}=\tfrac{CA'}{A'B}=k$. Prove that the ratio of the area of the triangle formed by the lines $AA',BB',CC'$ over the area of $\triangle ABC$ is $\tfrac{(k-1)^2}{(k^2+k+1)}$.

2013 Hitotsubashi University Entrance Examination, 2

Tags: geometry , geometry proposed

Given four points $O,\ A,\ B,\ C$ on a plane such that $OA=4,\ OB=3,\ OC=2,\ \overrightarrow{OB}\cdot \overrightarrow{OC}=3.$ Find the maximum area of $\triangle{ABC}$.

1996 All-Russian Olympiad, 2

Tags: geometry proposed , geometry

Several hikers travel at fixed speeds along a straight road. It is known that over some period of time, the sum of their pairwise distances is monotonically decreasing. Show that there is a hiker, the sum of whose distances to the other hikers is monotonically decreasing over the same period. [i]A. Shapovalov[/i]

1977 IMO Longlists, 58

Tags: inequalities , geometry , trigonometry , area of a triangle , Heron's formula , geometry proposed

Prove that for every triangle the following inequality holds: \[\frac{ab+bc+ca}{4S} \geq \cot \frac{\pi}{6}.\] where $a, b, c$ are lengths of the sides and $S$ is the area of the triangle.

2009 Indonesia TST, 4

Tags: geometry , geometry proposed

Let $ ABCD$ be a convex quadrilateral. Let $ M,N$ be the midpoints of $ AB,AD$ respectively. The foot of perpendicular from $ M$ to $ CD$ is $ K$, the foot of perpendicular from $ N$ to $ BC$ is $ L$. Show that if $ AC,BD,MK,NL$ are concurrent, then $ KLMN$ is a cyclic quadrilateral.

2013 Sharygin Geometry Olympiad, 23

Tags: symmetry , geometry proposed , geometry

Two convex polytopes $A$ and $B$ do not intersect. The polytope $A$ has exactly $2012$ planes of symmetry. What is the maximal number of symmetry planes of the union of $A$ and $B$, if $B$ has a) $2012$, b) $2013$ symmetry planes? c) What is the answer to the question of p.b), if the symmetry planes are replaced by the symmetry axes?

2012 Puerto Rico Team Selection Test, 3

Tags: geometry proposed , geometry

$ABC$ is a triangle that is inscribed in a circle. The angle bisectors of $A, B, C$ meet the circle at $D, E, F$, respectively. Show that $AD$ is perpendicular to $EF$.

2012 ITAMO, 5

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

$ABCD$ is a square. Describe the locus of points $P$, different from $A, B, C, D$, on that plane for which \[\widehat{APB}+\widehat{CPD}=180^\circ\]

2001 Brazil Team Selection Test, Problem 3

Tags: geometry , trapezoid , geometry proposed

In a triangle $ABC,$ the internal and external bisectors of the angle $A$ intersect the line $BC$ at $D$ and $E$ respectively. The line $AC$ meets the circle with diameter $DE$ again at $F.$ The tangent line to the circle $ABF$ at $A$ meets the circle with diameter $DE$ again at $G.$ Show that $AF = AG.$

1970 IMO Longlists, 46

Tags: geometry proposed , geometry

Given a triangle $ABC$ and a plane $\pi$ having no common points with the triangle, find a point $M$ such that the triangle determined by the points of intersection of the lines $MA,MB,MC$ with $\pi$ is congruent to the triangle $ABC$.

2014 Iran MO (3rd Round), 1

Tags: geometry , circumcircle , geometry proposed

In the circumcircle of triange $\triangle ABC,$ $AA'$ is a diameter. We draw lines $l'$ and $l$ from $A'$ parallel with Internal and external bisector of the vertex $A$. $l'$ Cut out $AB , BC$ at $B_1$ and $B_2$. $l$ Cut out $AC , BC$ at $C_1$ and $C_2$. Prove that the circumcircles of $\triangle ABC$ $\triangle CC_1C_2$ and $\triangle BB_1B_2$ have a common point. (20 points)

2002 Romania Team Selection Test, 1

Tags: symmetry , trigonometry , geometry proposed , geometry

Let $ABCDE$ be a cyclic pentagon inscribed in a circle of centre $O$ which has angles $\angle B=120^{\circ},\angle C=120^{\circ},$ $\angle D=130^{\circ},\angle E=100^{\circ}$. Show that the diagonals $BD$ and $CE$ meet at a point belonging to the diameter $AO$. [i]Dinu Șerbănescu[/i]

1985 IMO Longlists, 31

Tags: conics , ellipse , geometry proposed , geometry , concurrency

Let $E_1, E_2$, and $E_3$ be three mutually intersecting ellipses, all in the same plane. Their foci are respectively $F_2, F_3; F_3, F_1$; and $F_1, F_2$. The three foci are not on a straight line. Prove that the common chords of each pair of ellipses are concurrent.

2004 France Team Selection Test, 3

Tags: analytic geometry , inequalities , pigeonhole principle , triangle inequality , complex numbers , geometry proposed , geometry

Each point of the plane with two integer coordinates is the center of a disk with radius $ \frac {1} {1000}$. Prove that there exists an equilateral triangle whose vertices belong to distinct disks. Prove that such a triangle has side-length greater than 96.

2005 Iran MO (3rd Round), 2

Tags: geometry , circumcircle , Euler , power of a point , radical axis , geometry proposed

Suppose $O$ is circumcenter of triangle $ABC$. Suppose $\frac{S(OAB)+S(OAC)}2=S(OBC)$. Prove that the distance of $O$ (circumcenter) from the radical axis of the circumcircle and the 9-point circle is \[\frac {a^2}{\sqrt{9R^2-(a^2+b^2+c^2)}}\]

1980 IMO, 3

Tags: geometry proposed , geometry

Two circles $C_1$ and $C_2$ are tangent at a point $P$. The straight line at $D$ is tangent at $A$ to one of the circles and cuts the other circle at the points $B$ and $C$. Prove that the straight line $PA$ is a bisector (interior or exterior) of the angle $BPC$.

2014 Mexico National Olympiad, 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.

1998 Poland - Second Round, 2

Tags: geometry , geometry proposed

In triangle $ABC$, the angle $\angle BCA$ is obtuse and $\angle BAC = 2\angle ABC\,.$ The line through $B$ and perpendicular to $BC$ intersects line $AC$ in $D$. Let $M$ be the midpoint of $AB$. Prove that $\angle AMC=\angle BMD$. source : http://cage.ugent.be/~hvernaev/Olympiade/PMO982.pdf

2001 District Olympiad, 4

Tags: ratio , geometry proposed , geometry

Consider a convex qudrilateral $ABCD$ and $M\in (AB),\ N\in (CD)$ such that $\frac{AM}{BM}=\frac{DN}{CN}=k$. Prove that $BC\parallel AD$ if and only if \[MN=\frac{1}{k+1} AD+\frac{k}{k+1} BC\] [i]***[/i]

2009 Iran Team Selection Test, 5

Tags: geometry , angle bisector , geometry proposed

$ ABC$ is a triangle and $ AA'$ , $ BB'$ and $ CC'$ are three altitudes of this triangle . Let $ P$ be the feet of perpendicular from $ C'$ to $ A'B'$ , and $ Q$ is a point on $ A'B'$ such that $ QA \equal{} QB$ . Prove that : $ \angle PBQ \equal{} \angle PAQ \equal{} \angle PC'C$