Olympiad problems

2013 Bosnia Herzegovina Team Selection Test, 6

In triangle $ABC$, $I$ is the incenter. We have chosen points $P,Q,R$ on segments $IA,IB,IC$ respectively such that $IP\cdot IA=IQ \cdot IB=IR\cdot IC$. Prove that the points $I$ and $O$ belong to Euler line of triangle $PQR$ where $O$ is circumcenter of $ABC$.

2001 India IMO Training Camp, 1

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

If on $ \triangle ABC$, trinagles $ AEB$ and $ AFC$ are constructed externally such that $ \angle AEB\equal{}2 \alpha$, $ \angle AFB\equal{} 2 \beta$. $ AE\equal{}EB$, $ AF\equal{}FC$. COnstructed externally on $ BC$ is triangle $ BDC$ with $ \angle DBC\equal{} \beta$ , $ \angle BCD\equal{} \alpha$. Prove that 1. $ DA$ is perpendicular to $ EF$. 2. If $ T$ is the projection of $ D$ on $ BC$, then prove that $ \frac{DA}{EF}\equal{} 2 \frac{DT}{BC}$.

2002 India IMO Training Camp, 1

Tags: geometry , trapezoid , geometric transformation , homothety , power of a point , radical axis , geometry proposed

Let $A,B$ and $C$ be three points on a line with $B$ between $A$ and $C$. Let $\Gamma_1,\Gamma_2, \Gamma_3$ be semicircles, all on the same side of $AC$ and with $AC,AB,BC$ as diameters, respectively. Let $l$ be the line perpendicular to $AC$ through $B$. Let $\Gamma$ be the circle which is tangent to the line $l$, tangent to $\Gamma_1$ internally, and tangent to $\Gamma_3$ externally. Let $D$ be the point of contact of $\Gamma$ and $\Gamma_3$. The diameter of $\Gamma$ through $D$ meets $l$ in $E$. Show that $AB=DE$.

2003 China Team Selection Test, 1

Tags: geometry , trigonometry , calculus , geometry proposed

Let $ ABCD$ be a quadrilateral which has an incircle centered at $ O$. Prove that \[ OA\cdot OC\plus{}OB\cdot OD\equal{}\sqrt{AB\cdot BC\cdot CD\cdot DA}\]

2010 Canadian Mathematical Olympiad Qualification Repechage, 2

Tags: geometry , rectangle , geometry proposed

Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$, respectively, and meet perpendicularly at $T$. $Q$ is on $AT$, $S$ is on $BT$, and $R$ is on the circle, so that $QRST$ is a rectangle with $QT = 8$ and $ST = 9$. Determine the radius of the circle.

2005 Junior Balkan Team Selection Tests - Romania, 1

Tags: geometry , circumcircle , geometry proposed

Let $\mathcal{C}_1(O_1)$ and $\mathcal{C}_2(O_2)$ be two circles which intersect in the points $A$ and $B$. The tangent in $A$ at $\mathcal{C}_2$ intersects the circle $\mathcal{C}_1$ in $C$, and the tangent in $A$ at $\mathcal{C}_1$ intersects $\mathcal{C}_2$ in $D$. A ray starting from $A$ and lying inside the $\angle CAD$ intersects the circles $\mathcal{C}_1$, $\mathcal{C}_2$ in the points $M$ and $N$ respectively, and the circumcircle of $\triangle ACD$ in $P$. Prove that $AM=NP$.

2011 India National Olympiad, 1

Tags: trigonometry , symmetry , geometry proposed , geometry

Let $D,E,F$ be points on the sides $BC,CA,AB$ respectively of a triangle $ABC$ such that $BD=CE=AF$ and $\angle BDF=\angle CED=\angle AFE.$ Show that $\triangle ABC$ is equilateral.

2002 India IMO Training Camp, 1

Tags: geometry , trapezoid , geometric transformation , homothety , power of a point , radical axis , geometry proposed

Let $A,B$ and $C$ be three points on a line with $B$ between $A$ and $C$. Let $\Gamma_1,\Gamma_2, \Gamma_3$ be semicircles, all on the same side of $AC$ and with $AC,AB,BC$ as diameters, respectively. Let $l$ be the line perpendicular to $AC$ through $B$. Let $\Gamma$ be the circle which is tangent to the line $l$, tangent to $\Gamma_1$ internally, and tangent to $\Gamma_3$ externally. Let $D$ be the point of contact of $\Gamma$ and $\Gamma_3$. The diameter of $\Gamma$ through $D$ meets $l$ in $E$. Show that $AB=DE$.

2000 Korea - Final Round, 3

Tags: geometry , rectangle , perpendicular bisector , geometry proposed

A rectangle $ABCD$ is inscribed in a circle with centre $O$. The exterior bisectors of $\angle ABD$ and $\angle ADB$ intersect at $P$; those of $\angle DAB$ and $\angle DBA$ intersect at $Q$; those of $\angle ACD$ and $\angle ADC$ intersect at $R$; and those of $\angle DAC$ and $\angle DCA$ intersect at $S$. Prove that $P,Q,R$, and $S$ are concyclic.

2006 QEDMO 3rd, 6

Tags: geometry , geometric transformation , reflection , trapezoid , homothety , geometry proposed

The incircle of a triangle $ABC$ touches its sides $BC$, $CA$, $AB$ at the points $X$, $Y$, $Z$, respectively. Let $X^{\prime}$, $Y^{\prime}$, $Z^{\prime}$ be the reflections of these points $X$, $Y$, $Z$ in the external angle bisectors of the angles $CAB$, $ABC$, $BCA$, respectively. Show that $Y^{\prime}Z^{\prime}\parallel BC$, $Z^{\prime}X^{\prime}\parallel CA$ and $X^{\prime}Y^{\prime}\parallel AB$.

2007 Kazakhstan National Olympiad, 1

Tags: geometry , rhombus , function , geometry proposed

Convex quadrilateral $ABCD$ with $AB$ not equal to $DC$ is inscribed in a circle. Let $AKDL$ and $CMBN$ be rhombs with same side of $a$. Prove that the points $K, L, M, N$ lie on a circle.

2014 Iran Geometry Olympiad (senior), 5:

Tags: geometry , analytic geometry , graphing lines , slope , parallelogram , symmetry , geometry proposed

Two points $P$ and $Q$ lying on side $BC$ of triangle $ABC$ and their distance from the midpoint of $BC$ are equal.The perpendiculars from $P$ and $Q$ to $BC$ intersect $AC$ and $AB$ at $E$ and $F$,respectively.$M$ is point of intersection $PF$ and $EQ$.If $H_1$ and $H_2$ be the orthocenters of triangles $BFP$ and $CEQ$, respectively, prove that $ AM\perp H_1H_2 $. Author:Mehdi E'tesami Fard , Iran

1998 Iran MO (2nd round), 2

Tags: geometry , circumcircle , perpendicular bisector , cyclic quadrilateral , geometry proposed

Let $ABC$ be a triangle and $AB<AC<BC$. Let $D,E$ be points on the side $BC$ and the line $AB$, respectively ($A$ is between $B,E$) such that $BD=BE=AC$. The circumcircle of $\Delta BED$ meets the side $AC$ at $P$ and $BP$ meets the circumcircle of $\Delta ABC$ at $Q$. Prove that: \[ AQ+CQ=BP. \]

2007 Kazakhstan National Olympiad, 2

Tags: geometry , circumcircle , geometry proposed

Let $ABC$ be an isosceles triangle with $AC = BC$ and $I$ is the center of the inscribed circle. The point $P$ lies on the circle circumscribed about the triangle $AIB$ and lies inside the triangle $ABC$. Straight lines passing through point $P$ parallel to $CA$ and $CB$ intersect $AB$ at points $D$ and $E$, respectively. The line through $P$ which is parallel to $AB$ intersects $CA$ and $CB$ at points $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ meet at the circumcircle of triangle $ABC$.

2014 Moldova Team Selection Test, 3

Tags: geometry , circumcircle , trigonometry , geometry proposed

Let $\triangle ABC$ be a triangle with $\angle A$-acute. Let $P$ be a point inside $\triangle ABC$ such that $\angle BAP = \angle ACP$ and $\angle CAP =\angle ABP$. Let $M, N$ be the centers of the incircle of $\triangle ABP$ and $\triangle ACP$, and $R$ the radius of the circumscribed circle of $\triangle AMN$. Prove that $\displaystyle \frac{1}{R}=\frac{1}{AB}+\frac{1}{AC}+\frac{1}{AP}. $

2009 Romania Team Selection Test, 1

Tags: inequalities , geometry proposed , geometry

Let $ABCD$ be a circumscribed quadrilateral such that $AD>\max\{AB,BC,CD\}$, $M$ be the common point of $AB$ and $CD$ and $N$ be the common point of $AC$ and $BD$. Show that \[90^{\circ}<m(\angle AND)<90^{\circ}+\frac{1}{2}m(\angle AMD).\] Fixed, thank you Luis.

2005 Morocco TST, 4

Tags: geometry , power of a point , geometry proposed

Let $ABCD$ be a cyclic qudrilaterlal such that $AB.BC=2.CD.DA$ Prove that $8.BD^2 \leq 9.AC^2$

2023 Romania EGMO TST, P3

Tags: geometry , circumcircle , cyclic quadrilateral , geometry proposed

In a cyclic quadrilateral $ABCD$ with $AB=AD$ points $M$,$N$ lie on the sides $BC$ and $CD$ respectively so that $MN=BM+DN$ . Lines $AM$ and $AN$ meet the circumcircle of $ABCD$ again at points $P$ and $Q$ respectively. Prove that the orthocenter of the triangle $APQ$ lies on the segment $MN$ .

2006 Taiwan National Olympiad, 2

Tags: inequalities , geometry proposed , geometry

In triangle $ABC$, $D$ is the midpoint of side $AB$. $E$ and $F$ are points arbitrarily chosen on segments $AC$ and $BC$, respectively. Show that $[DEF] < [ADE] + [BDF]$.

2011 All-Russian Olympiad Regional Round, 11.6

Tags: geometry , circumcircle , perpendicular bisector , geometry proposed

$\omega$ is the circumcirle of an acute triangle $ABC$. The tangent line passing through $A$ intersects the tangent lines passing through points $B$ and $C$ at points $K$ and $L$, respectively. The line parallel to $AB$ through $K$ and the line parallel to $AC$ through $L$ intersect at point $P$. Prove that $BP=CP$. (Author: P. Kozhevnikov)

2010 Contests, 2

Tags: geometry , circumcircle , cyclic quadrilateral , geometry proposed

In a cyclic quadrilateral $ABCD$ with $AB=AD$ points $M$,$N$ lie on the sides $BC$ and $CD$ respectively so that $MN=BM+DN$ . Lines $AM$ and $AN$ meet the circumcircle of $ABCD$ again at points $P$ and $Q$ respectively. Prove that the orthocenter of the triangle $APQ$ lies on the segment $MN$ .

2008 Singapore Team Selection Test, 1

Tags: geometry proposed , geometry

In triangle $ABC$, $D$ is a point on $AB$ and $E$ is a point on $AC$ such that $BE$ and $CD$ are bisectors of $\angle B$ and $\angle C$ respectively. Let $Q,M$ and $N$ be the feet of perpendiculars from the midpoint $P$ of $DE$ onto $BC,AB$ and $AC$, respectively. Prove that $PQ=PM+PN$.

2007 Pre-Preparation Course Examination, 2

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

a) Prove that center of smallest sphere containing a finite subset of $\mathbb R^{n}$ is inside convex hull of the point that lie on sphere. b) $A$ is a finite subset of $\mathbb R^{n}$, and distance of every two points of $A$ is not larger than 1. Find radius of the largest sphere containing $A$.

2000 Brazil National Olympiad, 6

Tags: geometry , 3D geometry , geometry proposed

Let it be is a wooden unit cube. We cut along every plane which is perpendicular to the segment joining two distinct vertices and bisects it. How many pieces do we get?

2010 CentroAmerican, 6

Tags: ratio , geometry proposed , geometry

Let $\Gamma$ and $\Gamma_1$ be two circles internally tangent at $A$, with centers $O$ and $O_1$ and radii $r$ and $r_1$, respectively ($r>r_1$). $B$ is a point diametrically opposed to $A$ in $\Gamma$, and $C$ is a point on $\Gamma$ such that $BC$ is tangent to $\Gamma_1$ at $P$. Let $A'$ the midpoint of $BC$. Given that $O_1A'$ is parallel to $AP$, find the ratio $r/r_1$.