Olympiad problems

2006 Taiwan TST Round 1, 1

Circle $O$ is the incircle of the square $ABCD$. $O$ is tangent to $AB$ and $AD$ at $E$ and $F$, respectively. Let $K$ be a point on the minor arc $EF$, and let the tangent of $O$ at $K$ intersect $AB$, $AC$, $AD$ at $X$, $Y$, $Z$, respectively. Show that $\displaystyle \frac{AX}{XB} + \frac{AY}{YC} + \frac{AZ}{ZD} =1$.

2008 Korea - Final Round, 5

Tags: geometry , circumcircle , geometry proposed

Quadrilateral $ABCD$ is inscribed in a circle $O$. Let $AB\cap CD=E$ and $P\in BC, EP\perp BC$, $R\in AD, ER\perp AD$, $EP\cap AD=Q, ER\cap BC=S$ Let $K$ be the midpoint of $QS$ Prove that $E, K, O$ are collinear.

2005 ITAMO, 3

Tags: geometry , circumcircle , incenter , greatest common divisor , angle bisector , geometry proposed

Two circles $\gamma_1, \gamma_2$ in a plane, with centers $A$ and $B$ respectively, intersect at $C$ and $D$. Suppose that the circumcircle of $ABC$ intersects $\gamma_1$ in $E$ and $\gamma_2$ in $F$, where the arc $EF$ not containing $C$ lies outside $\gamma_1$ and $\gamma_2$. Prove that this arc $EF$ is bisected by the line $CD$.

1999 Iran MO (2nd round), 2

Tags: symmetry , vector , geometry proposed , geometry

$ABC$ is a triangle with $\angle{B}>45^{\circ}$ , $\angle{C}>45^{\circ}$. We draw the isosceles triangles $CAM,BAN$ on the sides $AC,AB$ and outside the triangle, respectively, such that $\angle{CAM}=\angle{BAN}=90^{\circ}$. And we draw isosceles triangle $BPC$ on the side $BC$ and inside the triangle such that $\angle{BPC}=90^{\circ}$. Prove that $\Delta{MPN}$ is an isosceles triangle, too, and $\angle{MPN}=90^{\circ}$.

2000 Mediterranean Mathematics Olympiad, 2

Tags: geometry proposed , geometry

Suppose that in the exterior of a convex quadrilateral $ABCD$ equilateral triangles $XAB,YBC,ZCD,WDA$ with centroids $S_1,S_2,S_3,S_4$ respectively are constructed. Prove that $S_1S_3\perp S_2S_4$ if and only if $AC=BD$.

2014 ELMO Shortlist, 9

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

Let $P$ be a point inside a triangle $ABC$ such that $\angle PAC= \angle PCB$. Let the projections of $P$ onto $BC$, $CA$, and $AB$ be $X,Y,Z$ respectively. Let $O$ be the circumcenter of $\triangle XYZ$, $H$ be the foot of the altitude from $B$ to $AC$, $N$ be the midpoint of $AC$, and $T$ be the point such that $TYPO$ is a parallelogram. Show that $\triangle THN$ is similar to $\triangle PBC$. [i]Proposed by Sammy Luo[/i]

2005 Germany Team Selection Test, 2

Tags: geometry , circumcircle , angle bisector , geometry proposed

Let $ABC$ be a triangle satisfying $BC < CA$. Let $P$ be an arbitrary point on the side $AB$ (different from $A$ and $B$), and let the line $CP$ meet the circumcircle of triangle $ABC$ at a point $S$ (apart from the point $C$). Let the circumcircle of triangle $ASP$ meet the line $CA$ at a point $R$ (apart from $A$), and let the circumcircle of triangle $BPS$ meet the line $CB$ at a point $Q$ (apart from $B$). Prove that the excircle of triangle $APR$ at the side $AP$ is identical with the excircle of triangle $PQB$ at the side $PQ$ if and only if the point $S$ is the midpoint of the arc $AB$ on the circumcircle of triangle $ABC$.

2000 JBMO ShortLists, 23

Tags: geometry proposed , geometry

The point $P$ is inside of an equilateral triangle with side length $10$ so that the distance from $P$ to two of the sides are $1$ and $3$. Find the distance from $P$ to the third side.

2010 Romania Team Selection Test, 3

Tags: geometry proposed , geometry

Let $\mathcal{L}$ be a finite collection of lines in the plane in general position (no two lines in $\mathcal{L}$ are parallel and no three are concurrent). Consider the open circular discs inscribed in the triangles enclosed by each triple of lines in $\mathcal{L}$. Determine the number of such discs intersected by no line in $\mathcal{L}$, in terms of $|\mathcal{L}|$. [i]B. Aronov et al.[/i]

2002 Iran Team Selection Test, 1

Tags: geometry , geometry proposed

$ABCD$ is a convex quadrilateral. We draw its diagnals to divide the quadrilateral to four triabgles. $P$ is the intersection of diagnals. $I_{1},I_{2},I_{3},I_{4}$ are excenters of $PAD,PAB,PBC,PCD$(excenters corresponding vertex $P$). Prove that $I_{1},I_{2},I_{3},I_{4}$ lie on a circle iff $ABCD$ is a tangential quadrilateral.

2010 Sharygin Geometry Olympiad, 4

Tags: geometry , circumcircle , cyclic quadrilateral , geometry proposed

The diagonals of a cyclic quadrilateral $ABCD$ meet in a point $N.$ The circumcircles of triangles $ANB$ and $CND$ intersect the sidelines $BC$ and $AD$ for the second time in points $A_1,B_1,C_1,D_1.$ Prove that the quadrilateral $A_1B_1C_1D_1$ is inscribed in a circle centered at $N.$

2013 Sharygin Geometry Olympiad, 2

Tags: inequalities , geometry proposed , geometry

Let $ABCD$ is a tangential quadrilateral such that $AB=CD>BC$. $AC$ meets $BD$ at $L$. Prove that $\widehat{ALB}$ is acute. [hide]According to the jury, they want to propose a more generalized problem is to prove $(AB-CD)^2 < (AD-BC)^2$, but this problem has appeared some time ago[/hide]

2003 Tournament Of Towns, 3

Tags: geometry , trapezoid , geometry proposed

Points $K$ and $L$ are chosen on the sides $AB$ and $BC$ of the isosceles $\triangle ABC$ ($AB = BC$) so that $AK +LC = KL$. A line parallel to $BC$ is drawn through midpoint $M$ of the segment $KL$, intersecting side $AC$ at point $N$. Find the value of $\angle KNL$.

1991 Hungary-Israel Binational, 2

Tags: geometry , inradius , perimeter , perpendicular bisector , geometry proposed

The vertices of a square sheet of paper are $ A$, $ B$, $ C$, $ D$. The sheet is folded in a way that the point $ D$ is mapped to the point $ D'$ on the side $ BC$. Let $ A'$ be the image of $ A$ after the folding, and let $ E$ be the intersection point of $ AB$ and $ A'D'$. Let $ r$ be the inradius of the triangle $ EBD'$. Prove that $ r\equal{}A'E$.

2007 Junior Balkan Team Selection Tests - Romania, 2

Tags: geometry , trapezoid , trigonometry , power of a point , radical axis , geometry proposed

Let $ABCD$ be a trapezium $(AB \parallel CD)$ and $M,N$ be the intersection points of the circles of diameters $AD$ and $BC$. Prove that $O \in MN$, where $O \in AC \cap BD$.

1999 Vietnam National Olympiad, 2

Tags: geometry , circumcircle , geometry proposed

let a triangle ABC and A',B',C' be the midpoints of the arcs BC,CA,AB respectively of its circumcircle. A'B',A'C' meets BC at $ A_1,A_2$ respectively. Pairs of point $ (B_1,B_2),(C_1,C_2)$ are similarly defined. Prove that $ A_1A_2 \equal{} B_1B_2 \equal{} C_1C_2$ if and only if triangle ABC is equilateral.

2008 Moldova Team Selection Test, 3

Tags: trigonometry , geometry proposed , geometry

In triangle $ ABC$ the bisector of $ \angle ACB$ intersects $ AB$ at $ D$. Consider an arbitrary circle $ O$ passing through $ C$ and $ D$, so that it is not tangent to $ BC$ or $ CA$. Let $ O\cap BC \equal{} \{M\}$ and $ O\cap CA \equal{} \{N\}$. a) Prove that there is a circle $ S$ so that $ DM$ and $ DN$ are tangent to $ S$ in $ M$ and $ N$, respectively. b) Circle $ S$ intersects lines $ BC$ and $ CA$ in $ P$ and $ Q$ respectively. Prove that the lengths of $ MP$ and $ NQ$ do not depend on the choice of circle $ O$.

2009 Olympic Revenge, 1

Tags: geometry , circumcircle , projective geometry , geometry proposed

Given a scalene triangle $ABC$ with circuncenter $O$ and circumscribed circle $\Gamma$. Let $D, E ,F$ the midpoints of $BC, AC, AB$. Let $M=OE \cap AD$, $N=OF \cap AD$ and $P=CM \cap BN$. Let $X=AO \cap PE$, $Y=AP \cap OF$. Let $r$ the tangent of $\Gamma$ through $A$. Prove that $r, EF, XY$ are concurrent.

2007 CentroAmerican, 3

Tags: geometry , parallelogram , incenter , perpendicular bisector , geometry proposed

Consider a circle $S$, and a point $P$ outside it. The tangent lines from $P$ meet $S$ at $A$ and $B$, respectively. Let $M$ be the midpoint of $AB$. The perpendicular bisector of $AM$ meets $S$ in a point $C$ lying inside the triangle $ABP$. $AC$ intersects $PM$ at $G$, and $PM$ meets $S$ in a point $D$ lying outside the triangle $ABP$. If $BD$ is parallel to $AC$, show that $G$ is the centroid of the triangle $ABP$. [i]Arnoldo Aguilar (El Salvador)[/i]

2003 Bulgaria Team Selection Test, 5

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

Let $ABCD$ be a circumscribed quadrilateral and let $P$ be the orthogonal projection of its in center on $AC$. Prove that $\angle {APB}=\angle {APD}$

2009 Macedonia National Olympiad, 2

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

Let $O$ be the centre of the incircle of $\triangle ABC$. Points $K,L$ are the intersection points of the circles circumscribed about triangles $BOC,AOC$ respectively with the bisectors of the angles at $A,B$ respectively $(K,L\not= O)$. Also $P$ is the midpoint of segment $KL$, $M$ is the reflection of $O$ with respect to $P$ and $N$ is the reflection of $O$ with respect to line $KL$. Prove that the points $K,L,M$ and $N$ lie on the same circle.

2014 Peru Iberoamerican Team Selection Test, P5

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

The incircle $\odot (I)$ of $\triangle ABC$ touch $AC$ and $AB$ at $E$ and $F$ respectively. Let $H$ be the foot of the altitude from $A$, if $R \equiv IC \cap AH, \ \ Q \equiv BI \cap AH$ prove that the midpoint of $AH$ lies on the radical axis between $\odot (REC)$ and $\odot (QFB)$ I hope that this is not repost :)

2020 Bundeswettbewerb Mathematik, 3

Tags: geometry , geometry proposed , circumcircle , cyclic quadrilateral , moving points

Two lines $m$ and $n$ intersect in a unique point $P$. A point $M$ moves along $m$ with constant speed, while another point $N$ moves along $n$ with the same speed. They both pass through the point $P$, but not at the same time. Show that there is a fixed point $Q \ne P$ such that the points $P,Q,M$ and $N$ lie on a common circle all the time.

1999 Baltic Way, 13

Tags: trigonometry , geometry , incenter , geometry proposed

The bisectors of the angles $A$ and $B$ of the triangle $ABC$ meet the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Assuming that $AE+BD=AB$, determine the angle $C$.

2007 Junior Balkan Team Selection Tests - Romania, 2

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

Let $w_{1}$ and $w_{2}$ be two circles which intersect at points $A$ and $B$. Consider $w_{3}$ another circle which cuts $w_{1}$ in $D,E$, and it is tangent to $w_{2}$ in the point $C$, and also tangent to $AB$ in $F$. Consider $G \in DE \cap AB$, and $H$ the symetric point of $F$ w.r.t $G$. Find $\angle{HCF}$.