Olympiad problems

Kvant 2024, M2800

Let $ABCD$ be a parallelogram. Let $M$ be the midpoint of the arc $AC$ containing $B$ of the circumcircle of $ABC$ . Let $E$ be a point on segment $AD$ and $F$ a point on segment $CD$ such that $ME=MD=MF$. Show that $BMEF$ is cyclic. [i]Proposed by A. Tereshin[/i]

2009 Ukraine National Mathematical Olympiad, 4

Tags: geometry , trapezoid , circumcircle , perpendicular bisector , geometry proposed

In the trapezoid $ABCD$ we know that $CD \perp BC, $ and $CD \perp AD .$ Circle $w$ with diameter $AB$ intersects $AD$ in points $A$ and $P,$ tangent from $P$ to $w$ intersects $CD$ at $M.$ The second tangent from $M$ to $w$ touches $w$ at $Q.$ Prove that midpoint of $CD$ lies on $BQ.$

2006 Greece National Olympiad, 3

Tags: ratio , IMO Shortlist , geometry proposed , geometry

Let a triangle $ABC$ and the cevians $AL, BN , CM$ such that $AL$ is the bisector of angle $A$. If $\angle ALB = \angle ANM$, prove that $\angle MNL = 90$.

1995 Bulgaria National Olympiad, 2

Tags: geometry , circumcircle , geometry proposed

Let triangle ABC has semiperimeter $ p$. E,F are located on AB such that $ CE\equal{}CF\equal{}p$. Prove that the C-excircle of triangle ABC touches the circumcircle (EFC).

2012 Iran Team Selection Test, 1

Tags: geometry , geometric transformation , reflection , perimeter , rotation , induction , geometry proposed

Consider a regular $2^k$-gon with center $O$ and label its sides clockwise by $l_1,l_2,...,l_{2^k}$. Reflect $O$ with respect to $l_1$, then reflect the resulting point with respect to $l_2$ and do this process until the last side. Prove that the distance between the final point and $O$ is less than the perimeter of the $2^k$-gon. [i]Proposed by Hesam Rajabzade[/i]

2008 Junior Balkan Team Selection Tests - Romania, 3

Tags: geometry , geometric transformation , homothety , trigonometry , ratio , geometry proposed

Let $ ABC$ be an acute-angled triangle. We consider the equilateral triangle $ A'UV$, where $ A' \in (BC)$, $ U\in (AC)$ and $ V\in(AB)$ such that $ UV \parallel BC$. We define the points $ B',C'$ in the same way. Prove that $ AA'$, $ BB'$ and $ CC'$ are concurrent.

2009 Middle European Mathematical Olympiad, 9

Tags: geometry , parallelogram , circumcircle , geometry proposed

Let $ ABCD$ be a parallelogram with $ \angle BAD \equal{} 60$ and denote by $ E$ the intersection of its diagonals. The circumcircle of triangle $ ACD$ meets the line $ BA$ at $ K \ne A$, the line $ BD$ at $ P \ne D$ and the line $ BC$ at $ L\ne C$. The line $ EP$ intersects the circumcircle of triangle $ CEL$ at points $ E$ and $ M$. Prove that triangles $ KLM$ and $ CAP$ are congruent.

2013 Turkey MO (2nd round), 3

Tags: geometry , rectangle , geometry proposed

Let $n$ be a positive integer and $P_1, P_2, \ldots, P_n$ be different points on the plane such that distances between them are all integers. Furthermore, we know that the distances $P_iP_1, P_iP_2, \ldots, P_iP_n$ forms the same sequence for all $i=1,2, \ldots, n$ when these numbers are arranged in a non-decreasing order. Find all possible values of $n$.

2006 Turkey Team Selection Test, 1

Tags: geometry , circumcircle , quadratics , trigonometry , Pythagorean Theorem , algebra , geometry proposed

Find the maximum value for the area of a heptagon with all vertices on a circle and two diagonals perpendicular.

2012 China Team Selection Test, 1

Tags: geometry , circumcircle , geometry proposed , Inversion

Given two circles ${\omega _1},{\omega _2}$, $S$ denotes all $\Delta ABC$ satisfies that ${\omega _1}$ is the circumcircle of $\Delta ABC$, ${\omega _2}$ is the $A$- excircle of $\Delta ABC$ , ${\omega _2}$ touches $BC,CA,AB$ at $D,E,F$. $S$ is not empty, prove that the centroid of $\Delta DEF$ is a fixed point.

1991 IberoAmerican, 6

Tags: geometry , circumcircle , quadratics , parallelogram , algebra , geometry proposed

Let $M$, $N$ and $P$ be three non-collinear points. Construct using straight edge and compass a triangle for which $M$ and $N$ are the midpoints of two of its sides, and $P$ is its orthocenter.

2010 Contests, 1

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

$D, \: E , \: F$ are points on the sides $AB, \: BC, \: CA,$ respectively, of a triangle $ABC$ such that $AD=AF, \: BD=BE,$ and $DE=DF.$ Let $I$ be the incenter of the triangle $ABC,$ and let $K$ be the point of intersection of the line $BI$ and the tangent line through $A$ to the circumcircle of the triangle $ABI.$ Show that $AK=EK$ if $AK=AD.$

2005 India IMO Training Camp, 1

Tags: Euler , geometry , circumcircle , incenter , geometry proposed

Let $ABC$ be a triangle with all angles $\leq 120^{\circ}$. Let $F$ be the Fermat point of triangle $ABC$, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of the three triangles $BFC$, $CFA$ and $AFB$, draw its Euler line - that is, the line connecting its circumcenter and its centroid. Prove that these three Euler lines pass through one common point. [i]Remark.[/i] The Fermat point $F$ is also known as the [b]first Fermat point[/b] or the [b]first Toricelli point[/b] of triangle $ABC$. [i]Floor van Lamoen[/i]

2008 Bosnia Herzegovina Team Selection Test, 1

Tags: inequalities , trigonometry , calculus , derivative , geometry , geometry proposed

Prove that in an isosceles triangle $ \triangle ABC$ with $ AC\equal{}BC\equal{}b$ following inequality holds $ b> \pi r$, where $ r$ is inradius.

2007 Indonesia MO, 7

Tags: geometry , circumcircle , geometry proposed

Points $ A,B,C,D$ are on circle $ S$, such that $ AB$ is the diameter of $ S$, but $ CD$ is not the diameter. Given also that $ C$ and $ D$ are on different sides of $ AB$. The tangents of $ S$ at $ C$ and $ D$ intersect at $ P$. Points $ Q$ and $ R$ are the intersections of line $ AC$ with line $ BD$ and line $ AD$ with line $ BC$, respectively. (a) Prove that $ P$, $ Q$, and $ R$ are collinear. (b) Prove that $ QR$ is perpendicular to line $ AB$.

2014 Dutch BxMO/EGMO TST, 3

Tags: geometry proposed , geometry

In triangle $ABC$, $I$ is the centre of the incircle. There is a circle tangent to $AI$ at $I$ which passes through $B$. This circle intersects $AB$ once more in $P$ and intersects $BC$ once more in $Q$. The line $QI$ intersects $AC$ in $R$. Prove that $|AR|\cdot |BQ|=|P I|^2$

2014 Regional Competition For Advanced Students, 4

Tags: geometry proposed , geometry

For a point $P$ in the interior of a triangle $ABC$ let $D$ be the intersection of $AP$ with $BC$, let $E$ be the intersection of $BP$ with $AC$ and let $F$ be the intersection of $CP$ with $AB$.Furthermore let $Q$ and $R$ be the intersections of the parallel to $AB$ through $P$ with the sides $AC$ and $BC$, respectively. Likewise, let $S$ and $T$ be the intersections of the parallel to $BC$ through $P$ with the sides $AB$ and $AC$, respectively.In a given triangle $ABC$, determine all points $P$ for which the triangles $PRD$, $PEQ$and $PTE$ have the same area.

2011 Iran MO (3rd Round), 1

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

We have $4$ circles in plane such that any two of them are tangent to each other. we connect the tangency point of two circles to the tangency point of two other circles. Prove that these three lines are concurrent. [i]proposed by Masoud Nourbakhsh[/i]

2010 Romania National Olympiad, 1

Tags: geometry , incenter , geometry proposed

In a triangle $ABC$ denote by $D,E,F$ the points where the angle bisectors of $\angle CAB,\angle ABC,\angle BCA$ respectively meet it's circumcircle. a) Prove that the orthocenter of triangle $DEF$ coincides with the incentre of triangle $ABC$. b) Prove that if $\overrightarrow{AD}+\overrightarrow{BE}+\overrightarrow{CF}=0$, then the triangle $ABC$ is equilateral. [i]Marin Ionescu[/i]

2000 Brazil National Olympiad, 1

Tags: geometry , geometric transformation , reflection , geometry proposed

A rectangular piece of paper has top edge $AD$. A line $L$ from $A$ to the bottom edge makes an angle $x$ with the line $AD$. We want to trisect $x$. We take $B$ and $C$ on the vertical ege through $A$ such that $AB = BC$. We then fold the paper so that $C$ goes to a point $C'$ on the line $L$ and $A$ goes to a point $A'$ on the horizontal line through $B$. The fold takes $B$ to $B'$. Show that $AA'$ and $AB'$ are the required trisectors.

2014 All-Russian Olympiad, 2

Tags: geometry , trapezoid , symmetry , geometry proposed

Let $ABCD$ be a trapezoid with $AB\parallel CD$ and $ \Omega $ is a circle passing through $A,B,C,D$. Let $ \omega $ be the circle passing through $C,D$ and intersecting with $CA,CB$ at $A_1$, $B_1$ respectively. $A_2$ and $B_2$ are the points symmetric to $A_1$ and $B_1$ respectively, with respect to the midpoints of $CA$ and $CB$. Prove that the points $A,B,A_2,B_2$ are concyclic. [i]I. Bogdanov[/i]

2012 Turkey MO (2nd round), 2

Tags: trigonometry , geometry , circumcircle , projective geometry , geometry proposed

Let $ABC$ be a isosceles triangle with $AB=AC$ an $D$ be the foot of perpendicular of $A$. $P$ be an interior point of triangle $ADC$ such that $m(APB)>90$ and $m(PBD)+m(PAD)=m(PCB)$. $CP$ and $AD$ intersects at $Q$, $BP$ and $AD$ intersects at $R$. Let $T$ be a point on $[AB]$ and $S$ be a point on $[AP$ and not belongs to $[AP]$ satisfying $m(TRB)=m(DQC)$ and $m(PSR)=2m(PAR)$. Show that $RS=RT$

2014 Taiwan TST Round 1, 4

Tags: geometry proposed , geometry

Let $ABC$ be an acute triangle and let $D$ be the foot of the $A$-bisector. Moreover, let $M$ be the midpoint of $AD$. The circle $\omega_1$ with diameter $AC$ meets $BM$ at $E$, while the circle $\omega_2$ with diameter $AB$ meets $CM$ at $F$. Assume that $E$ and $F$ lie inside $ABC$. Prove that $B$, $E$, $F$, $C$ are concyclic.

2017 Taiwan TST Round 2, 2

Tags: geometry , geometry proposed

Given a $ \triangle ABC $ and three points $ D, E, F $ such that $ DB = DC, $ $ EC = EA, $ $ FA = FB, $ $ \measuredangle BDC = \measuredangle CEA = \measuredangle AFB. $ Let $ \Omega_D $ be the circle with center $ D $ passing through $ B, C $ and similarly for $ \Omega_E, \Omega_F. $ Prove that the radical center of $ \Omega_D, \Omega_E, \Omega_F $ lies on the Euler line of $ \triangle DEF. $ [i]Proposed by Telv Cohl[/i]

2008 Iran Team Selection Test, 9

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

$ I_a$ is the excenter of the triangle $ ABC$ with respect to $ A$, and $ AI_a$ intersects the circumcircle of $ ABC$ at $ T$. Let $ X$ be a point on $ TI_a$ such that $ XI_a^2\equal{}XA.XT$. Draw a perpendicular line from $ X$ to $ BC$ so that it intersects $ BC$ in $ A'$. Define $ B'$ and $ C'$ in the same way. Prove that $ AA'$, $ BB'$ and $ CC'$ are concurrent.