Olympiad problems

2009 Sharygin Geometry Olympiad, 6

Find the locus of excenters of right triangles with given hypotenuse.

2007 Romania Team Selection Test, 4

Tags: function , geometry , geometric transformation , reflection , geometry proposed

Let $\mathcal O_{1}$ and $\mathcal O_{2}$ two exterior circles. Let $A$, $B$, $C$ be points on $\mathcal O_{1}$ and $D$, $E$, $F$ points on $\mathcal O_{1}$ such that $AD$ and $BE$ are the common exterior tangents to these two circles and $CF$ is one of the interior tangents to these two circles, and such that $C$, $F$ are in the interior of the quadrilateral $ABED$. If $CO_{1}\cap AB=\{M\}$ and $FO_{2}\cap DE=\{N\}$ then prove that $MN$ passes through the middle of $CF$.

2007 Indonesia TST, 1

Tags: geometry , circumcircle , incenter , geometry proposed

Given triangle $ ABC$ and its circumcircle $ \Gamma$, let $ M$ and $ N$ be the midpoints of arcs $ BC$ (that does not contain $ A$) and $ CA$ (that does not contain $ B$), repsectively. Let $ X$ be a variable point on arc $ AB$ that does not contain $ C$. Let $ O_1$ and $ O_2$ be the incenter of triangle $ XAC$ and $ XBC$, respectively. Let the circumcircle of triangle $ XO_1O_2$ meets $ \Gamma$ at $ Q$. (a) Prove that $ QNO_1$ and $ QMO_2$ are similar. (b) Find the locus of $ Q$ as $ X$ varies.

2011 Tokyo Instutute Of Technology Entrance Examination, 2

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

For a positive real number $t$, in the coordiante space, consider 4 points $O(0,\ 0,\ 0),\ A(t,\ 0,\ 0),\ B(0,\ 1,\ 0),\ C(0,\ 0,\ 1)$. Let $r$ be the radius of the sphere $P$ which is inscribed to all faces of the tetrahedron $OABC$. When $t$ moves, find the maximum value of $\frac{\text{vol[P]}}{\text{vol[OABC]}}.$

2000 Federal Competition For Advanced Students, Part 2, 1

Tags: Euler , geometry , circumcircle , angle bisector , geometry proposed

In a non-equilateral acute-angled triangle $ABC$ with $\angle C = 60^\circ$, $U$ is the circumcenter, $H$ the orthocenter and $D$ the intersection of $AH$ and $BC$. Prove that the Euler line $HU$ bisects the angle $BHD$.

2007 All-Russian Olympiad, 7

Tags: geometry , 3D geometry , tetrahedron , inequalities , geometry proposed

Given a tetrahedron $ T$. Valentin wants to find two its edges $ a,b$ with no common vertices so that $ T$ is covered by balls with diameters $ a,b$. Can he always find such a pair? [i]A. Zaslavsky[/i]

2003 All-Russian Olympiad, 2

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

Two circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$ respectively intersect at $A$ and $B$. The tangents at $A$ to $S_1$ and $S_2$ meet segments $BO_2$ and $BO_1$ at $K$ and $L$ respectively. Show that $KL \parallel O_1O_2.$

2005 Taiwan TST Round 1, 2

Tags: trigonometry , geometry , incenter , symmetry , cyclic quadrilateral , geometry proposed

$P$ is a point in the interior of $\triangle ABC$, and $\angle ABP = \angle PCB = 10^\circ$. (a) If $\angle PBC = 10^\circ$ and $\angle ACP = 20^\circ$, what is the value of $\angle BAP$? (b) If $\angle PBC = 20^\circ$ and $\angle ACP = 10^\circ$, what is the value of $\angle BAP$?

1973 Bundeswettbewerb Mathematik, 2

Tags: geometry proposed , geometry

In a planar lake, every point can be reached by a straight line from the point $A$. The same holds for the point $B$. Show that this holds for every point on the segment $[AB]$, too.

2002 Iran Team Selection Test, 7

Tags: geometry , 3D geometry , sphere , circumcircle , power of a point , radical axis , geometry proposed

$S_{1},S_{2},S_{3}$ are three spheres in $\mathbb R^{3}$ that their centers are not collinear. $k\leq8$ is the number of planes that touch three spheres. $A_{i},B_{i},C_{i}$ is the point that $i$-th plane touch the spheres $S_{1},S_{2},S_{3}$. Let $O_{i}$ be circumcenter of $A_{i}B_{i}C_{i}$. Prove that $O_{i}$ are collinear.

2024 Bundeswettbewerb Mathematik, 3

Tags: geometry , circumcircle , geometry proposed , Angle Chasing , parallelogram

Let $ABCD$ be a parallelogram whose diagonals intersect in $M$. Suppose that the circumcircle of $ABM$ intersects the segment $AD$ in a point $E \ne A$ and that the circumcircle of $EMD$ intersects the segment $BE$ in a point $F \ne E$. Show that $\angle ACB=\angle DCF$.

2007 Indonesia MO, 1

Tags: geometry , perpendicular bisector , geometry proposed

Let $ ABC$ be a triangle with $ \angle ABC\equal{}\angle ACB\equal{}70^{\circ}$. Let point $ D$ on side $ BC$ such that $ AD$ is the altitude, point $ E$ on side $ AB$ such that $ \angle ACE\equal{}10^{\circ}$, and point $ F$ is the intersection of $ AD$ and $ CE$. Prove that $ CF\equal{}BC$.

2005 Morocco TST, 4

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

Consider a cyclic quadrilateral $ABCD$, and let $S$ be the intersection of $AC$ and $BD$. Let $E$ and $F$ the orthogonal projections of $S$ on $AB$ and $CD$ respectively. Prove that the perpendicular bisector of segment $EF$ meets the segments $AD$ and $BC$ at their midpoints.

2015 Iran Geometry Olympiad, 1

Tags: geometry proposed , geometry

Given a circle and Points $P,B,A$ on it.Point $Q$ is Interior of this circle such that: $1)$ $\angle PAQ=90$. $ 2)PQ=BQ$. Prove that $\angle AQB - \angle PQA=\stackrel{\frown}{AB}$. proposed by Davoud Vakili, Iran.

1997 Vietnam National Olympiad, 1

Tags: geometry , perimeter , geometry proposed

Given a circle (O,R). A point P lies inside the circle, OP=d, d<R. We consider quadrilaterals ABCD, inscribed in (O), such that AC is perp to BD at point P. Evaluate the maximum and minimum values of the perimeter of ABCD in terms of R and d.

2006 JBMO ShortLists, 11

Tags: geometry proposed , geometry

Circles $ \mathcal{C}_1$ and $ \mathcal{C}_2$ intersect at $ A$ and $ B$. Let $ M\in AB$. A line through $ M$ (different from $ AB$) cuts circles $ \mathcal{C}_1$ and $ \mathcal{C}_2$ at $ Z,D,E,C$ respectively such that $ D,E\in ZC$. Perpendiculars at $ B$ to the lines $ EB,ZB$ and $ AD$ respectively cut circle $ \mathcal{C}_2$ in $ F,K$ and $ N$. Prove that $ KF\equal{}NC$.

1993 All-Russian Olympiad Regional Round, 10.7

Tags: geometry , parallelogram , ratio , geometry proposed

Points $ M,N$ are taken on sides $ BC,CD$ respectively of parallelogram $ ABCD$. Let $ E\equal{}BD\cap AM, F\equal{}BD\cap AN$. Diagonal $ BD$ cuts triangle $ AMN$ into two parts. Prove that these two parts have equal area if and only if the point $ K$ given by $ EK\parallel{}AD, FK\parallel{}AB$ lies on segment $ MN$.

1989 Balkan MO, 3

Tags: inequalities , geometry proposed , geometry

Let $G$ be the centroid of a triangle $ABC$ and let $d$ be a line that intersects $AB$ and $AC$ at $B_{1}$ and $C_{1}$, respectively, such that the points $A$ and $G$ are not separated by $d$. Prove that: $[BB_{1}GC_{1}]+[CC_{1}GB_{1}] \geq \frac{4}{9}[ABC]$.

2004 Iran MO (3rd Round), 8

Tags: geometry , perimeter , geometry proposed

$\mathbb{P}$ is a n-gon with sides $l_1 ,...,l_n$ and vertices on a circle. Prove that no n-gon with this sides has area more than $\mathbb{P}$

2013 Sharygin Geometry Olympiad, 1

Tags: geometry proposed , geometry

Let $ABC$ be an isosceles triangle with $AB = BC$. Point $E$ lies on the side $AB$, and $ED$ is the perpendicular from $E$ to $BC$. It is known that $AE = DE$. Find $\angle DAC$.

2007 China Western Mathematical Olympiad, 3

Tags: geometry , circumcircle , geometric transformation , homothety , projective geometry , similar triangles , geometry proposed

Let $ P$ be an interior point of an acute angled triangle $ ABC$. The lines $ AP,BP,CP$ meet $ BC,CA,AB$ at points $ D,E,F$ respectively. Given that triangle $ \triangle DEF$ and $ \triangle ABC$ are similar, prove that $ P$ is the centroid of $ \triangle ABC$.

2000 Iran MO (3rd Round), 3

Tags: vector , geometry proposed , geometry

Two triangles $ ABC$and $ A'B'C'$ are positioned in the space such that the length of every side of $ \triangle ABC$ is not less than $ a$, and the length of every side of $ \triangle A'B'C'$ is not less than $ a'$. Prove that one can select a vertex of $ \triangle ABC$ and a vertex of $ \triangle A'B'C'$ so that the distance between the two selected vertices is not less than $ \sqrt {\frac {a^2 \plus{} a'^2}{3}}$.

2006 Korea - Final Round, 1

Tags: geometry , geometry proposed

In a triangle $ABC$ with $AB\not = AC$, the incircle touches the sides $BC, CA, AB$ at $D, E, F$ , respectively. Line $AD$ meets the incircle again at $P$ . The line $EF$ and the line through $P$ perpendicular to $AD$ meet at $Q$. Line $AQ$ intersects $DE$ at $X$ and $DF$ at $Y$ . Prove that $A$ is the midpoint of $XY$.

2013 Sharygin Geometry Olympiad, 22

Tags: geometry proposed , geometry

The common perpendiculars to the opposite sidelines of a nonplanar quadrilateral are mutually orthogonal. Prove that they intersect.

2002 Iran MO (3rd Round), 5

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

$\omega$ is circumcirlce of triangle $ABC$. We draw a line parallel to $BC$ that intersects $AB,AC$ at $E,F$ and intersects $\omega$ at $U,V$. Assume that $M$ is midpoint of $BC$. Let $\omega'$ be circumcircle of $UMV$. We know that $R(ABC)=R(UMV)$. $ME$ and $\omega'$ intersect at $T$, and $FT$ intersects $\omega'$ at $S$. Prove that $EF$ is tangent to circumcircle of $MCS$.