Olympiad problems

2014 Korea National Olympiad, 1

There is a convex quadrilateral $ ABCD $ which satisfies $ \angle A=\angle D $. Let the midpoints of $ AB, AD, CD $ be $ L,M,N $. Let's say the intersection point of $ AC, BD $ be $ E $ . Let's say point $ F $ which lies on $ \overrightarrow{ME} $ satisfies $ \overline{ME}\times \overline{MF}=\overline{MA}^{2} $. Prove that $ \angle LFM=\angle MFN $. :)

1977 IMO Longlists, 17

Tags: geometry proposed , geometry

A ball $K$ of radius $r$ is touched from the outside by mutually equal balls of radius $R$. Two of these balls are tangent to each other. Moreover, for two balls $K_1$ and $K_2$ tangent to $K$ and tangent to each other there exist two other balls tangent to $K_1,K_2$ and also to $K$. How many balls are tangent to $K$? For a given $r$ determine $R$.

2024 All-Russian Olympiad, 4

Tags: geometry , cyclic quadrilateral , geometry proposed

A quadrilateral $ABCD$ without parallel sides is inscribed in a circle $\omega$. We draw a line $\ell_a \parallel BC$ through the point $A$, a line $\ell_b \parallel CD$ through the point $B$, a line $\ell_c \parallel DA$ through the point $C$, and a line $\ell_d \parallel AB$ through the point $D$. Suppose that the quadrilateral whose successive sides lie on these four straight lines is inscribed in a circle $\gamma$ and that $\omega$ and $\gamma$ intersect in points $E$ and $F$. Show that the lines $AC, BD$ and $EF$ intersect in one point. [i]Proposed by A. Kuznetsov[/i]

2001 JBMO ShortLists, 8

Tags: analytic geometry , geometry proposed , geometry

Prove that no three points with integer coordinates can be the vertices of an equilateral triangle.

2007 Junior Balkan Team Selection Tests - Romania, 1

Tags: geometry proposed , geometry

Let $ABC$ a triangle and $M,N,P$ points on $AB,BC$, respective $CA$, such that the quadrilateral $CPMN$ is a paralelogram. Denote $R \in AN \cap MP$, $S \in BP \cap MN$, and $Q \in AN \cap BP$. Prove that $[MRQS]=[NQP]$.

Cono Sur Shortlist - geometry, 2012.G1

Tags: geometry , circumcircle , geometry proposed

Let $ABCD$ be a cyclic quadrilateral. Let $P$ be the intersection of $BC$ and $AD$. Line $AC$ intersects the circumcircle of triangle $BDP$ in points $S$ and $T$, with $S$ between $A$ and $C$. Line $BD$ intersects the circumcircle of triangle $ACP$ in points $U$ and $V$, with $U$ between $B$ and $D$. Prove that $PS$ = $PT$ = $PU$ = $PV$.

2009 JBMO Shortlist, 4

Tags: geometry , trapezoid , geometry proposed , pentagon

Let $ ABCDE$ be a convex pentagon such that $ AB\plus{}CD\equal{}BC\plus{}DE$ and $ k$ a circle with center on side $ AE$ that touches the sides $ AB$, $ BC$, $ CD$ and $ DE$ at points $ P$, $ Q$, $ R$ and $ S$ (different from vertices of the pentagon) respectively. Prove that lines $ PS$ and $ AE$ are parallel.

1998 Baltic Way, 13

Tags: geometry proposed , geometry

In convex pentagon $ABCDE$, the sides $AE,BC$ are parallel and $\angle ADE=\angle BDC$. The diagonals $AC$ and $BE$ intersect at $P$. Prove that $\angle EAD=\angle BDP$ and $\angle CBD=\angle ADP$.

2013 Korea - Final Round, 4

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

For a triangle $ ABC $, let $ B_1 ,C_1 $ be the excenters of $ B, C $. Line $B_1 C_1 $ meets with the circumcircle of $ \triangle ABC $ at point $ D (\ne A) $. $ E $ is the point which satisfies $ B_1 E \bot CA $ and $ C_1 E \bot AB $. Let $ w $ be the circumcircle of $ \triangle ADE $. The tangent to the circle $ w $ at $ D $ meets $ AE $ at $ F $. $ G , H $ are the points on $ AE, w $ such that $ DGH \bot AE $. The circumcircle of $ \triangle HGF $ meets $ w $ at point $ I ( \ne H ) $, and $ J $ be the foot of perpendicular from $ D $ to $ AH $. Prove that $ AI $ passes the midpoint of $ DJ $.

2001 District Olympiad, 3

Tags: symmetry , geometry , parallelogram , geometry proposed

Consider a triangle $\Delta ABC$ and three points $D,E,F$ such that: $B$ and $E$ are on different side of the line $AC$, $C$ and $D$ are on different sides of $AB$, $A$ and $F$ are on the same side of the line $BC$. Also $\Delta ADB \sim \Delta CEA \sim \Delta CFB$. Let $M$ be the middle point of $AF$. Prove that: a)$\Delta BDF \sim \Delta FEC$. b) $M$ is the middle point of $DE$. [i]Dan Branzei[/i]

2011 All-Russian Olympiad Regional Round, 9.6

Tags: geometry , circumcircle , perpendicular bisector , geometry proposed

Initially, there are three different points on the plane. Every minute, three points are chosen, for example $A$, $B$ and $C$, and a new point $D$ is generated which is symmetric to $A$ with respect to the perpendicular bisector of line segment $BC$. 24 hours later, it turns out that among all the points that were generated, there exist three collinear points. Prove that the three initial points were also collinear. (Author: V. Shmarov)

2002 Iran MO (3rd Round), 7

Tags: trigonometry , geometry proposed , geometry

In triangle $ABC$, $AD$ is angle bisector ($D$ is on $BC$) if $AB+AD=CD$ and $AC+AD=BC$, what are the angles of $ABC$?

2012 India National Olympiad, 1

Tags: geometry , inequalities , trigonometry , function , circumcircle , topology , geometry proposed

Let $ABCD$ be a quadrilateral inscribed in a circle. Suppose $AB=\sqrt{2+\sqrt{2}}$ and $AB$ subtends $135$ degrees at center of circle . Find the maximum possible area of $ABCD$.

2002 Olympic Revenge, 2

Tags: olympic revenge 2002 , geometry proposed , circles , geometry , circumcircle

$ABCD$ is a inscribed quadrilateral. $P$ is the intersection point of its diagonals. $O$ is its circumcenter. $\Gamma$ is the circumcircle of $ABO$. $\Delta$ is the circumcircle of $CDO$. $M$ is the midpoint of arc $AB$ on $\Gamma$ who doesn't contain $O$. $N$ is the midpoint of arc $CD$ on $\Delta$ who doesn't contain $O$. Show that $M,N,P$ are collinear.

2009 China Western Mathematical Olympiad, 2

Tags: geometry , circumcircle , geometry proposed

Given an acute triangle $ABC$, $D$ is a point on $BC$. A circle with diameter $BD$ intersects line $AB,AD$ at $X,P$ respectively (different from $B,D$).The circle with diameter $CD$ intersects $AC,AD$ at $Y,Q$ respectively (different from $C,D$). Draw two lines through $A$ perpendicular to $PX,QY$, the feet are $M,N$ respectively.Prove that $\triangle AMN$ is similar to $\triangle ABC$ if and only if $AD$ passes through the circumcenter of $\triangle ABC$.

2024 Polish MO Finals, 1

Tags: geometry , geometry proposed , rectangle , angles

Let $X$ be an interior point of a rectangle $ABCD$. Let the bisectors of $\angle DAX$ and $\angle CBX$ intersect in $P$. A point $Q$ satisfies $\angle QAP=\angle QBP=90^\circ$. Show that $PX=QX$.

2006 Grigore Moisil Intercounty, 1

Tags: trigonometry , geometry , similar triangles , geometry proposed

Let $ABC$ be a triangle with $b\neq c$. Points $D$ is the midpoint of $BC$ and let $E$ be the foot of angle $A$ bisector. In the exterior of the triangle we construct the similar triangles $AMB$ and $ANC$ . Prove: a) $MN\bot AD \Longleftrightarrow MA \bot AB$ b) $MN\bot AE \Longleftrightarrow M,A,N$ are colinear.

2019 Taiwan TST Round 1, 6

Tags: geometry proposed , geometry

Given a triangle $ \triangle ABC $. Denote its incenter and orthocenter by $ I, H $, respectively. If there is a point $ K $ with $$ AH+AK = BH+BK = CH+CK $$ Show that $ H, I, K $ are collinear. [i]Proposed by Evan Chen[/i]

1995 Dutch Mathematical Olympiad, 2

Tags: geometry , trapezoid , geometry proposed

For any point $ P$ on a segment $ AB$, isosceles and right-angled triangles $ AQP$ and $ PRB$ are constructed on the same side of $ AB$, with $ AP$ and $ PB$ as the bases. Determine the locus of the midpoint $ M$ of $ QR$ when $ P$ describes the segment $ AB$.

2008 Bosnia Herzegovina Team Selection Test, 2

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

Let $ AD$ be height of triangle $ \triangle ABC$ and $ R$ circumradius. Denote by $ E$ and $ F$ feet of perpendiculars from point $ D$ to sides $ AB$ and $ AC$. If $ AD\equal{}R\sqrt{2}$, prove that circumcenter of triangle $ \triangle ABC$ lies on line $ EF$.

2006 Iran Team Selection Test, 3

Tags: geometry , geometry proposed

Suppose $ABC$ is a triangle with $M$ the midpoint of $BC$. Suppose that $AM$ intersects the incircle at $K,L$. We draw parallel line from $K$ and $L$ to $BC$ and name their second intersection point with incircle $X$ and $Y$. Suppose that $AX$ and $AY$ intersect $BC$ at $P$ and $Q$. Prove that $BP=CQ$.

1993 Dutch Mathematical Olympiad, 2

Tags: ratio , geometry proposed , geometry

In a triangle $ ABC$ with $ \angle A\equal{}90^{\circ}$, $ D$ is the midpoint of $ BC$, $ F$ that of $ AB$, $ E$ that of $ AF$ and $ G$ that of $ FB$. Segment $ AD$ intersects $ CE,CF$ and $ CG$ in $ P,Q$ and $ R$, respectively. Determine the ratio: $ \frac{PQ}{QR}$.

1997 Turkey Team Selection Test, 1

Tags: geometry , inradius , incenter , area of a triangle , geometry proposed

In a triangle $ABC$ with a right angle at $A$, $H$ is the foot of the altitude from $A$. Prove that the sum of the inradii of the triangles $ABC$, $ABH$, and $AHC$ is equal to $AH$.

2012 Serbia National Math Olympiad, 1

Tags: geometry , parallelogram , circumcircle , trigonometry , geometry proposed

Let $ABCD$ be a parallelogram and $P$ be a point on diagonal $BD$ such that $\angle PCB=\angle ACD$. Circumcircle of triangle $ABD$ intersects line $AC$ at points $A$ and $E$. Prove that \[\angle AED=\angle PEB.\]

2010 Contests, 3

Tags: geometry , circumcircle , incenter , projective geometry , geometry proposed

The circle $ \Gamma $ is inscribed to the scalene triangle $ABC$. $ \Gamma $ is tangent to the sides $BC, CA$ and $AB$ at $D, E$ and $F$ respectively. The line $EF$ intersects the line $BC$ at $G$. The circle of diameter $GD$ intersects $ \Gamma $ in $R$ ($ R\neq D $). Let $P$, $Q$ ($ P\neq R , Q\neq R $) be the intersections of $ \Gamma $ with $BR$ and $CR$, respectively. The lines $BQ$ and $CP$ intersects at $X$. The circumcircle of $CDE$ meets $QR$ at $M$, and the circumcircle of $BDF$ meet $PR$ at $N$. Prove that $PM$, $QN$ and $RX$ are concurrent. [i]Author: Arnoldo Aguilar, El Salvador[/i]