Olympiad problems

2009 Sharygin Geometry Olympiad, 14

Given triangle $ ABC$ of area 1. Let $ BM$ be the perpendicular from $ B$ to the bisector of angle $ C$. Determine the area of triangle $ AMC$.

2006 Junior Balkan Team Selection Tests - Moldova, 1

Tags: geometry proposed , geometry

Five segments have lengths such that any three of them can be sides of a - possibly degenerate - triangle. Also, the lengths of these segments are nonzero and pairwisely different. Prove that there exists at least one acute-angled triangle among these triangles.

2016 JBMO TST - Turkey, 5

Tags: geometry proposed , perpendicular lines , geometry

In an acute triangle $ABC$, the feet of the perpendiculars from $A$ and $C$ to the opposite sides are $D$ and $E$, respectively. The line passing through $E$ and parallel to $BC$ intersects $AC$ at $F$, the line passing through $D$ and parallel to $AB$ intersects $AC$ at $G$. The feet of the perpendiculars from $F$ to $DG$ and $GE$ are $K$ and $L$, respectively. $KL$ intersects $ED$ at $M$. Prove that $FM \perp ED$.

2010 Korea National Olympiad, 2

Tags: geometry , geometric transformation , homothety , circumcircle , ratio , projective geometry , geometry proposed

Let $ ABCD$ be a cyclic convex quadrilateral. Let $ E $ be the intersection of lines $ AB, CD $. $ P $ is the intersection of line passing $ B $ and perpendicular to $ AC $, and line passing $ C $ and perpendicular to $ BD$. $ Q $ is the intersection of line passing $ D $ and perpendicular to $ AC $, and line passing $ A $ and perpendicular to $ BD $. Prove that three points $ E, P, Q $ are collinear.

2014 Canada National Olympiad, 4

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

The quadrilateral $ABCD$ is inscribed in a circle. The point $P$ lies in the interior of $ABCD$, and $\angle P AB = \angle P BC = \angle P CD = \angle P DA$. The lines $AD$ and $BC$ meet at $Q$, and the lines $AB$ and $CD$ meet at $R$. Prove that the lines $P Q$ and $P R$ form the same angle as the diagonals of $ABCD$.

2004 Italy TST, 1

Tags: geometry proposed , geometry

Two circles $\gamma_1$ and $\gamma_2$ intersect at $A$ and $B$. A line $r$ through $B$ meets $\gamma_1$ at $C$ and $\gamma_2$ at $D$ so that $B$ is between $C$ and $D$. Let $s$ be the line parallel to $AD$ which is tangent to $\gamma_1$ at $E$, at the smaller distance from $AD$. Line $EA$ meets $\gamma_2$ in $F$. Let $t$ be the tangent to $\gamma_2$ at $F$. $(a)$ Prove that $t$ is parallel to $AC$. $(b)$ Prove that the lines $r,s,t$ are concurrent.

2021 German National Olympiad, 2

Tags: geometry , perimeter , geometry proposed , cyclic quadrilateral , interior

Let $P$ on $AB$, $Q$ on $BC$, $R$ on $CD$ and $S$ on $AD$ be points on the sides of a convex quadrilateral $ABCD$. Show that the following are equivalent: (1) There is a choice of $P,Q,R,S$, for which all of them are interior points of their side, such that $PQRS$ has minimal perimeter. (2) $ABCD$ is a cyclic quadrilateral with circumcenter in its interior.

2006 Switzerland Team Selection Test, 1

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

In the triangle $A,B,C$, let $D$ be the middle of $BC$ and $E$ the projection of $C$ on $AD$. Suppose $\angle ACE = \angle ABC$. Show that the triangle $ABC$ is isosceles or rectangle.

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?

2014 Kazakhstan National Olympiad, 1

Tags: geometry , circumcircle , geometry proposed

Given a scalene triangle $ABC$. Incircle of $\triangle{ABC{}}$ touches the sides $AB$ and $BC$ at points $C_1$ and $A_1$ respectively, and excircle of $\triangle{ABC}$ (on side $AC$) touches $AB$ and $BC$ at points $ C_2$ and $A_2$ respectively. $BN$ is bisector of $\angle{ABC}$ ($N$ lies on $BC$). Lines $A_1C_1$ and $A_2C_2$ intersects the line $AC$ at points $K_1$ and $K_2$ respectively. Let circumcircles of $\triangle{BK_1N}$ and $\triangle{BK_2N}$ intersect circumcircle of a $\triangle{ABC}$ at points $P_1$ and $P_2$ respectively. Prove that $AP_1$=$CP_2$

2010 Sharygin Geometry Olympiad, 17

Tags: geometry proposed , geometry

Construct a triangle, if the lengths of the bisectrix and of the altitude from one vertex, and of the median from another vertex are given.

2016 APMC, 2

Tags: geometry , geometry proposed

Let $ABC$ be a triangle with incenter $I$, and suppose that $AI$, $BI$, and $CI$ intersect $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. Let the circumcircles of $BDF$ and $CDE$ intersect at $D$ and $P$, and let $H$ be the orthocenter of $DEF$. Prove that $HI=HP$.

2005 Taiwan TST Round 2, 1

Tags: geometry proposed , geometry

It is known that there exists a point $P$ within the interior of $\triangle ABC$ satisfying the following conditions: (i) $\angle PAB \ge 30^\circ$ and $\angle APB \ge \angle PCB + 30^\circ$; (ii) $BP \cdot BC=CP \cdot AB.$ Prove that $\angle BAC \ge 60^\circ$, and that equality holds only when $\triangle ABC$ is equilateral.

2002 Polish MO Finals, 2

Tags: geometry , 3D geometry , sphere , geometry proposed

There is given a triangle $ABC$ in a space. A sphere does not intersect the plane of $ABC$. There are $4$ points $K, L, M, P$ on the sphere such that $AK, BL, CM$ are tangent to the sphere and $\frac{AK}{AP} = \frac{BL}{BP} = \frac{CM}{CP}$. Show that the sphere touches the circumsphere of $ABCP$.

2006 Junior Balkan Team Selection Tests - Romania, 2

Tags: complex numbers , geometry proposed , geometry

Let $ABC$ be a triangle and $A_1$, $B_1$, $C_1$ the midpoints of the sides $BC$, $CA$ and $AB$ respectively. Prove that if $M$ is a point in the plane of the triangle such that \[ \frac{MA}{MA_1} = \frac{MB}{MB_1} = \frac{MC}{MC_1} = 2 , \] then $M$ is the centroid of the triangle.

2009 Ukraine National Mathematical Olympiad, 3

Tags: geometry , circumcircle , parallelogram , cyclic quadrilateral , geometric transformation , geometry proposed

In triangle $ABC$ points $M, N$ are midpoints of $BC, CA$ respectively. Point $P$ is inside $ABC$ such that $\angle BAP = \angle PCA = \angle MAC .$ Prove that $\angle PNA = \angle AMB .$

2017 Bulgaria JBMO TST, 2

Tags: geometry , geometry proposed , junior geometry

Let $k$ be the incircle of triangle $ABC$. It touches $AB=c, BC=a, AC=b$ at $C_1, A_1, B_1$, respectively. Suppose that $KC_1$ is a diameter of the incircle. Let $C_1A_1$ intersect $KB_1$ at $N$ and $C_1B_1$ intersect $KA_1$ at $M$. Find the length of $MN$.

2015 Iran Geometry Olympiad, 3

Tags: geometry , geometry proposed

In triangle $ABC$ ,$M,N,K$ are midpoints of sides $BC,AC,AB$,respectively.Construct two semicircles with diameter $AB,AC$ outside of triangle $ABC$.$MK,MN$ intersect with semicircles in $X,Y$.The tangents to semicircles at $X,Y$ intersect at point $Z$.Prove that $AZ \perp BC$.(Mehdi E'tesami Fard)

2010 Kazakhstan National Olympiad, 1

Tags: geometry , conics , ellipse , rectangle , geometry proposed

Triangle $ABC$ is given. Consider ellipse $ \Omega _1$, passes through $C$ with focuses in $A$ and $B$. Similarly define ellipses $ \Omega _2 , \Omega _3$ with focuses $B,C$ and $C,A$ respectively. Prove, that if all ellipses have common point $D$ then $A,B,C,D$ lies on the circle. Ellipse with focuses $X,Y$, passes through $Z$- locus of point $T$, such that $XT+YT=XZ+YZ$

JBMO Geometry Collection, 2001

Tags: geometry , trigonometry , geometry proposed

Let $ABC$ be an equilateral triangle and $D$, $E$ points on the sides $[AB]$ and $[AC]$ respectively. If $DF$, $EF$ (with $F\in AE$, $G\in AD$) are the interior angle bisectors of the angles of the triangle $ADE$, prove that the sum of the areas of the triangles $DEF$ and $DEG$ is at most equal with the area of the triangle $ABC$. When does the equality hold? [i]Greece[/i]

2010 Romania Team Selection Test, 2

Tags: ratio , geometry proposed , geometry

Let $\ell$ be a line, and let $\gamma$ and $\gamma'$ be two circles. The line $\ell$ meets $\gamma$ at points $A$ and $B$, and $\gamma'$ at points $A'$ and $B'$. The tangents to $\gamma$ at $A$ and $B$ meet at point $C$, and the tangents to $\gamma'$ at $A'$ and $B'$ meet at point $C'$. The lines $\ell$ and $CC'$ meet at point $P$. Let $\lambda$ be a variable line through $P$ and let $X$ be one of the points where $\lambda$ meets $\gamma$, and $X'$ be one of the points where $\lambda$ meets $\gamma'$. Prove that the point of intersection of the lines $CX$ and $C'X'$ lies on a fixed circle. [i]Gazeta Matematica[/i]

2012 Tuymaada Olympiad, 2

Tags: geometry , rectangle , symmetry , angle bisector , geometry proposed

A rectangle $ABCD$ is given. Segment $DK$ is equal to $BD$ and lies on the half-line $DC$. $M$ is the midpoint of $BK$. Prove that $AM$ is the angle bisector of $\angle BAC$. [i]Proposed by S. Berlov[/i]

2012 Iran MO (3rd Round), 5

Tags: conics , parabola , geometry proposed , geometry

Two fixed lines $l_1$ and $l_2$ are perpendicular to each other at a point $Y$. Points $X$ and $O$ are on $l_2$ and both are on one side of line $l_1$. We draw the circle $\omega$ with center $O$ and radius $OY$. A variable point $Z$ is on line $l_1$. Line $OZ$ cuts circle $\omega$ in $P$. Parallel to $XP$ from $O$ intersects $XZ$ in $S$. Find the locus of the point $S$. [i]Proposed by Nima Hamidi[/i]

1994 Turkey Team Selection Test, 1

Tags: ratio , geometry , geometry proposed

Let $P,Q,R$ be points on the sides of $\triangle ABC$ such that $P \in [AB],Q\in[BC],R\in[CA]$ and $\frac{|AP|}{|AB|} = \frac {|BQ|}{|BC|} =\frac{|CR|}{|CA|} =k < \frac 12$ If $G$ is the centroid of $\triangle ABC$, find the ratio $\frac{Area(\triangle PQG)}{Area(\triangle PQR)}$ .

2002 Turkey MO (2nd round), 2

Tags: geometry , incenter , circumcircle , ratio , geometry proposed

Let $ABC$ be a triangle, and points $D,E$ are on $BA,CA$ respectively such that $DB=BC=CE$. Let $O,I$ be the circumcenter, incenter of $\triangle ABC$. Prove that the circumradius of $\triangle ADE$ is equal to $OI$.