Olympiad problems

2008 Turkey Team Selection Test, 1

In an $ ABC$ triangle such that $ m(\angle B)>m(\angle C)$, the internal and external bisectors of vertice $ A$ intersects $ BC$ respectively at points $ D$ and $ E$. $ P$ is a variable point on $ EA$ such that $ A$ is on $ [EP]$. $ DP$ intersects $ AC$ at $ M$ and $ ME$ intersects $ AD$ at $ Q$. Prove that all $ PQ$ lines have a common point as $ P$ varies.

2000 Italy TST, 2

Tags: linear algebra , matrix , symmetry , geometry , angle bisector , geometry unsolved

Let $ ABC$ be an isosceles right triangle and $M$ be the midpoint of its hypotenuse $AB$. Points $D$ and $E$ are taken on the legs $AC$ and $BC$ respectively such that $AD=2DC$ and $BE=2EC$. Lines $AE$ and $DM$ intersect at $F$. Show that $FC$ bisects the $\angle DFE$.

2005 China Girls Math Olympiad, 1

Tags: geometry , circumcircle , perpendicular bisector , geometry unsolved

As shown in the following figure, point $ P$ lies on the circumcicle of triangle $ ABC.$ Lines $ AB$ and $ CP$ meet at $ E,$ and lines $ AC$ and $ BP$ meet at $ F.$ The perpendicular bisector of line segment $ AB$ meets line segment $ AC$ at $ K,$ and the perpendicular bisector of line segment $ AC$ meets line segment $ AB$ at $ J.$ Prove that \[ \left(\frac{CE}{BF} \right)^2 \equal{} \frac{AJ \cdot JE}{AK \cdot KF}.\]

1986 IMO Longlists, 29

Tags: vector , geometry , geometric transformation , rotation , complex numbers , geometry unsolved

We define a binary operation $\star$ in the plane as follows: Given two points $A$ and $B$ in the plane, $C = A \star B$ is the third vertex of the equilateral triangle ABC oriented positively. What is the relative position of three points $I, M, O$ in the plane if $I \star (M \star O) = (O \star I)\star M$ holds?

2006 MOP Homework, 6

Tags: geometry unsolved , geometry

Suppose there are $18$ light houses on the Mexican gulf. Each of the lighthouses lightens an angle with size $20$ degrees. Prove that we can choose the directions of the lighthouses such that the whole gulf is lightened.

2008 Sharygin Geometry Olympiad, 8

Tags: geometry , perimeter , geometry unsolved

(B.Frenkin, A.Zaslavsky) A convex quadrilateral was drawn on the blackboard. Boris marked the centers of four excircles each touching one side of the quadrilateral and the extensions of two adjacent sides. After this, Alexey erased the quadrilateral. Can Boris define its perimeter?

2008 ISI B.Stat Entrance Exam, 4

Tags: geometry unsolved , geometry

Suppose $P$ and $Q$ are the centres of two disjoint circles $C_1$ and $C_2$ respectively, such that $P$ lies outside $C_2$ and $Q$ lies outside $C_1$. Two tangents are drawn from the point $P$ to the circle $C_2$, which intersect the circle $C_1$ at point $A$ and $B$. Similarly, two tangents are drawn from the point $Q$ to the circle $C_1$, which intersect the circle $C_2$ at points $M$ and $N$. Show that $AB=MN$

2000 Turkey Team Selection Test, 1

Tags: geometry , 3D geometry , prism , geometry unsolved

Show that any triangular prism of inﬁnite length can be cut by a plane such that the resulting intersection is an equilateral triangle.

2004 Nordic, 4

Tags: inequalities , geometry , circumcircle , inradius , geometry unsolved

Let $a, b, c$ be the sides and $R$ be the circumradius of a triangle. Prove that \[\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\ge\frac{1}{R^2}.\]

2013 Polish MO Finals, 4

Tags: geometry , 3D geometry , tetrahedron , rotation , trapezoid , geometry unsolved

Given is a tetrahedron $ABCD$ in which $AB=CD$ and the sum of measures of the angles $BAD$ and $BCD$ equals $180$ degrees. Prove that the measure of the angle $BAD$ is larger than the measure of the angle $ADC$.

Indonesia MO Shortlist - geometry, g6.2

Tags: geometry , circumcircle , projective geometry , geometry unsolved

Given an acute triangle $ABC$ with $AC>BC$ and the circumcenter of triangle $ABC$ is $O$. The altitude of triangle $ABC$ from $C$ intersects $AB$ and the circumcircle at $D$ and $E$, respectively. A line which passed through $O$ which is parallel to $AB$ intersects $AC$ at $F$. Show that the line $CO$, the line which passed through $F$ and perpendicular to $AC$, and the line which passed through $E$ and parallel with $DO$ are concurrent. [i]Fajar Yuliawan, Bandung[/i]

1996 Iran MO (3rd Round), 2

Tags: geometry , parallelogram , geometric transformation , rotation , geometry unsolved

Let $ABCD$ be a parallelogram. Construct the equilateral triangle $DCE$ on the side $DC$ and outside of parallelogram. Let $P$ be an arbitrary point in plane of $ABCD$. Show that \[PA+PB+AD \geq PE.\]

2008 Sharygin Geometry Olympiad, 4

Tags: symmetry , geometry , parallelogram , angle bisector , geometry unsolved

(F.Nilov, A.Zaslavsky) Let $ CC_0$ be a median of triangle $ ABC$; the perpendicular bisectors to $ AC$ and $ BC$ intersect $ CC_0$ in points $ A'$, $ B'$; $ C_1$ is the meet of lines $ AA'$ and $ BB'$. Prove that $ \angle C_1CA \equal{} \angle C_0CB$.

2004 South africa National Olympiad, 4

Tags: geometry , circumcircle , geometry unsolved

Let $A_1$ and $B_1$ be two points on the base $AB$ of isosceles triangle $ABC$ (with $\widehat{C}>60^\circ$) such that $\widehat{A_1CB_1}=\widehat{BAC}$. A circle externally tangent to the circumcircle of triangle $\triangle A_1B_1C$ is tangent also to rays $CA$ and $CB$ at points $A_2$ and $B_2$ respectively. Prove that $A_2B_2=2AB$.

2000 JBMO ShortLists, 17

Tags: geometry , geometry proposed , geometry unsolved

A triangle $ABC$ is given. Find all the pairs of points $X,Y$ so that $X$ is on the sides of the triangle, $Y$ is inside the triangle, and four non-intersecting segments from the set $\{XY, AX, AY, BX,BY, CX, CY\}$ divide the triangle $ABC$ into four triangles with equal areas.

2010 Postal Coaching, 6

Tags: geometry , trigonometry , geometry unsolved

Let $a,b,c$ denote the sides of a triangle and $[ABC]$ the area of the triangle as usual. $(a)$ If $6[ABC] = 2a^2+bc$, determine $A,B,C$. $(b)$ For all triangles, prove that $3a^2+3b^2 - c^2 \ge 4 \sqrt{3} [ABC]$.

1983 Dutch Mathematical Olympiad, 1

Tags: geometry unsolved , geometry

A triangle $ ABC$ can be divided into two isosceles triangles by a line through $ A$. Given that one of the angles of the triangles is $ 30^{\circ}$, find all possible values of the other two angles.

2005 MOP Homework, 6

Tags: geometry , circumcircle , incenter , exterior angle , angle bisector , geometry unsolved

A circle which is tangent to sides $AB$ and $BC$ of triangle $ABC$ is also tangent to its circumcircle at point $T$. If $I$ in the incenter of triangle $ABC$, show that $\angle ATI=\angle CTI$.

2014 Kurschak Competition, 2

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

We are given an acute triangle $ABC$, and inside it a point $P$, which is not on any of the heights $AA_1$, $BB_1$, $CC_1$. The rays $AP$, $BP$, $CP$ intersect the circumcircle of $ABC$ at points $A_2$, $B_2$, $C_2$. Prove that the circles $AA_1A_2$, $BB_1B_2$ and $CC_1C_2$ are concurrent.

1993 Turkey MO (2nd round), 2

Tags: geometry , parallelogram , geometry unsolved

I centered incircle of triangle $ABC$ $(m(\hat{B})=90^\circ)$ touches $\left[AB\right], \left[BC\right], \left[AC\right]$ respectively at $F, D, E$. $\left[CI\right]\cap\left[EF\right]={L}$ and $\left[DL\right]\cap\left[AB\right]=N$. Prove that $\left[AI\right]=\left[ND\right]$.

2011 Kazakhstan National Olympiad, 5

Tags: geometry , power of a point , geometry unsolved

Given a non-degenerate triangle $ABC$, let $A_{1}, B_{1}, C_{1}$ be the point of tangency of the incircle with the sides $BC, AC, AB$. Let $Q$ and $L$ be the intersection of the segment $AA_{1}$ with the incircle and the segment $B_{1}C_{1}$ respectively. Let $M$ be the midpoint of $B_{1}C_{1}$. Let $T$ be the point of intersection of $BC$ and $B_{1}C_{1}$. Let $P$ be the foot of the perpendicular from the point $L$ on the line $AT$. Prove that the points $A_{1}, M, Q, P$ lie on a circle.

2009 Polish MO Finals, 5

Tags: geometry , 3D geometry , sphere , tetrahedron , circumcircle , geometry unsolved

A sphere is inscribed in tetrahedron $ ABCD$ and is tangent to faces $ BCD,CAD,ABD,ABC$ at points $ P,Q,R,S$ respectively. Segment $ PT$ is the sphere's diameter, and lines $ TA,TQ,TR,TS$ meet the plane $ BCD$ at points $ A',Q',R',S'$. respectively. Show that $ A$ is the center of a circumcircle on the triangle $ S'Q'R'$.

2012 International Zhautykov Olympiad, 2

Tags: complex numbers , geometry unsolved , geometry

Equilateral triangles $ACB'$ and $BDC'$ are drawn on the diagonals of a convex quadrilateral $ABCD$ so that $B$ and $B'$ are on the same side of $AC$, and $C$ and $C'$ are on the same sides of $BD$. Find $\angle BAD + \angle CDA$ if $B'C' = AB+CD$.

1996 Irish Math Olympiad, 4

Tags: geometry , geometry unsolved

Let $ F$ be the midpoint of the side $ BC$ of a triangle $ ABC$. Isosceles right-angled triangles $ ABD$ and $ ACE$ are constructed externally on $ AB$ and $ AC$ with the right angles at $ D$ and $ E$. Prove that the triangle $ DEF$ is right-angled and isosceles.

2003 China Team Selection Test, 2

Tags: geometry , ratio , inequalities , analytic geometry , linear algebra , matrix , geometry unsolved

In triangle $ABC$, the medians and bisectors corresponding to sides $BC$, $CA$, $AB$ are $m_a$, $m_b$, $m_c$ and $w_a$, $w_b$, $w_c$ respectively. $P=w_a \cap m_b$, $Q=w_b \cap m_c$, $R=w_c \cap m_a$. Denote the areas of triangle $ABC$ and $PQR$ by $F_1$ and $F_2$ respectively. Find the least positive constant $m$ such that $\frac{F_1}{F_2}<m$ holds for any $\triangle{ABC}$.