Olympiad problems

2014 JBMO TST - Turkey, 1

In a triangle $ABC$, the external bisector of $\angle BAC$ intersects the ray $BC$ at $D$. The feet of the perpendiculars from $B$ and $C$ to line $AD$ are $E$ and $F$, respectively and the foot of the perpendicular from $D$ to $AC$ is $G$. Show that $\angle DGE + \angle DGF = 180^{\circ}$.

2010 ELMO Shortlist, 3

Tags: geometry , incenter , circumcircle , conics , parabola , geometry proposed

A circle $\omega$ not passing through any vertex of $\triangle ABC$ intersects each of the segments $AB$, $BC$, $CA$ in 2 distinct points. Prove that the incenter of $\triangle ABC$ lies inside $\omega$. [i]Evan O' Dorney.[/i]

2010 Indonesia TST, 1

Tags: geometry , trapezoid , circumcircle , geometry proposed

Let $ ABCD$ be a trapezoid such that $ AB \parallel CD$ and assume that there are points $ E$ on the line outside the segment $ BC$ and $ F$ on the segment $ AD$ such that $ \angle DAE \equal{} \angle CBF$. Let $ I,J,K$ respectively be the intersection of line $ EF$ and line $ CD$, the intersection of line $ EF$ and line $ AB$, and the midpoint of segment $ EF$. Prove that $ K$ is on the circumcircle of triangle $ CDJ$ if and only if $ I$ is on the circumcircle of triangle $ ABK$. [i]Utari Wijayanti, Bandung[/i]

1999 Baltic Way, 12

Tags: geometry , incenter , angle bisector , geometry proposed

In a triangle $ABC$ it is given that $2AB=AC+BC$. Prove that the incentre of $\triangle ABC$, the circumcentre of $\triangle ABC$, and the midpoints of $AC$ and $BC$ are concyclic.

2010 Contests, 3

Tags: geometry , geometric transformation , rotation , geometry proposed

Prove that there exists a set $S$ of lines in the three dimensional space satisfying the following conditions: $i)$ For each point $P$ in the space, there exist a unique line of $S$ containing $P$. $ii)$ There are no two lines of $S$ which are parallel.

2011 China Second Round Olympiad, 1

Tags: geometry , circumcircle , symmetry , cyclic quadrilateral , geometry proposed

Let $P,Q$ be the midpoints of diagonals $AC,BD$ in cyclic quadrilateral $ABCD$. If $\angle BPA=\angle DPA$, prove that $\angle AQB=\angle CQB$.

2005 Bulgaria Team Selection Test, 1

Tags: trigonometry , geometry proposed , geometry

Let $ABC$ be an acute triangle. Find the locus of the points $M$, in the interior of $\bigtriangleup ABC$, such that $AB-FG= \frac{MF.AG+MG.BF}{CM}$, where $F$ and $G$ are the feet of the perpendiculars from $M$ to the lines $BC$ and $AC$, respectively.

2004 Czech and Slovak Olympiad III A, 5

Tags: geometry , circumcircle , geometry proposed

Let $L$ be an arbitrary point on the minor arc $CD$ of the circumcircle of square $ABCD$. Let $K,M,N$ be the intersection points of $AL,CD$; $CL,AD$; and $MK,BC$ respectively. Prove that $B,M,L,N$ are concyclic.

2001 Romania National Olympiad, 4

Tags: geometry proposed , geometry

Consider the acute angle $ABC$. On the half-line $BC$ we consider the distinct points $P$ and $Q$ whose projections onto the line $AB$ are the points $M$ and $N$. Knowing that $AP=AQ$ and $AM^2-AN^2=BN^2-BM^2$, find the angle $ABC$.

1993 Irish Math Olympiad, 3

Tags: geometry proposed , geometry

A line $ l$ is tangent to a circle $ S$ at $ A$. For any points $ B,C$ on $ l$ on opposite sides of $ A$, let the other tangents from $ B$ and $ C$ to $ S$ intersect at a point $ P$. If $ B,C$ vary on $ l$ so that the product $ AB \cdot AC$ is constant, find the locus of $ P$.

2005 Poland - Second Round, 2

Tags: geometry , rhombus , trapezoid , geometry proposed

A rhombus $ABCD$ with $\angle BAD=60^{\circ}$ is given. Points $E$ on side $AB$ and $F$ on side $AD$ are such that $\angle ECF=\angle ABD$. Lines $CE$ and $CF$ respectively meet line $BD$ at $P$ and $Q$. Prove that $\frac{PQ}{EF}=\frac{AB}{BD}$.

2008 Iran MO (3rd Round), 1

Tags: geometry , circumcircle , geometric transformation , reflection , incenter , cyclic quadrilateral , geometry proposed

Let $ ABC$ be a triangle with $ BC > AC > AB$. Let $ A',B',C'$ be feet of perpendiculars from $ A,B,C$ to $ BC,AC,AB$, such that $ AA' \equal{} BB' \equal{} CC' \equal{} x$. Prove that: a) If $ ABC\sim A'B'C'$ then $ x \equal{} 2r$ b) Prove that if $ A',B'$ and $ C'$ are collinear, then $ x \equal{} R \plus{} d$ or $ x \equal{} R \minus{} d$. (In this problem $ R$ is the radius of circumcircle, $ r$ is radius of incircle and $ d \equal{} OI$)

2001 Junior Balkan MO, 3

Tags: geometry , trigonometry , geometry proposed

Let $ABC$ be an equilateral triangle and $D$, $E$ points on the sides $[AB]$ and $[AC]$ respectively. If $DF$, $EF$ (with $F\in AE$, $G\in AD$) are the interior angle bisectors of the angles of the triangle $ADE$, prove that the sum of the areas of the triangles $DEF$ and $DEG$ is at most equal with the area of the triangle $ABC$. When does the equality hold? [i]Greece[/i]

2003 China National Olympiad, 1

Tags: geometry , incenter , circumcircle , geometry proposed

Let $I$ and $H$ be the incentre and orthocentre of triangle $ABC$ respectively. Let $P,Q$ be the midpoints of $AB,AC$. The rays $PI,QI$ intersect $AC,AB$ at $R,S$ respectively. Suppose that $T$ is the circumcentre of triangle $BHC$. Let $RS$ intersect $BC$ at $K$. Prove that $A,I$ and $T$ are collinear if and only if $[BKS]=[CKR]$. [i]Shen Wunxuan[/i]

2010 Contests, 2

Tags: geometry , circumcircle , rotation , analytic geometry , perpendicular bisector , geometry proposed , LaLaLa

$AB$ is a diameter of a circle with center $O$. Let $C$ and $D$ be two different points on the circle on the same side of $AB$, and the lines tangent to the circle at points $C$ and $D$ meet at $E$. Segments $AD$ and $BC$ meet at $F$. Lines $EF$ and $AB$ meet at $M$. Prove that $E,C,M$ and $D$ are concyclic.

1986 IMO Longlists, 56

Tags: geometry proposed , geometry

Let $A_1A_2A_3A_4A_5A_6$ be a hexagon inscribed into a circle with center $O$. Consider the circular arc with endpoints $A_1,A_6$ not containing $A_2$. For any point $M$ of that arc denote by $h_i$ the distance from $M$ to the line $A_iA_{i+1} \ (1 \leq i \leq 5)$. Construct $M$ such that the sum $h_1 + \cdots + h_5$ is maximal.

2005 Taiwan TST Round 1, 1

Tags: geometry , trigonometry , ratio , geometric sequence , geometry proposed

Consider a circle $O_1$ with radius $R$ and a point $A$ outside the circle. It is known that $\angle BAC=60^\circ$, where $AB$ and $AC$ are tangent to $O_1$. We construct infinitely many circles $O_i$ $(i=1,2,\dots\>)$ such that for $i>1$, $O_i$ is tangent to $O_{i-1}$ and $O_{i+1}$, that they share the same tangent lines $AB$ and $AC$ with respect to $A$, and that none of the $O_i$ are larger than $O_1$. Find the total area of these circles. I know this problem was easy, but it still appeared in the TST, and so I posted it. It was kind of a disappointment for me.

2013 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: geometry proposed , geometry

Convex quadrilateral is divided by diagonals into four triangles with congruent inscribed circles. Prove that this quadrilateral is rhombus.

2007 Iran MO (3rd Round), 7

Tags: geometry , topology , 3D geometry , sphere , rotation , geometry proposed

A ring is the area between two circles with the same center, and width of a ring is the difference between the radii of two circles. [img]http://i18.tinypic.com/6cdmvi8.png[/img] a) Can we put uncountable disjoint rings of width 1(not necessarily same) in the space such that each two of them can not be separated. [img]http://i19.tinypic.com/4qgx30j.png[/img] b) What's the answer if 1 is replaced with 0?

2009 Sharygin Geometry Olympiad, 5

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

Given triangle $ ABC$. Point $ O$ is the center of the excircle touching the side $ BC$. Point $ O_1$ is the reflection of $ O$ in $ BC$. Determine angle $ A$ if $ O_1$ lies on the circumcircle of $ ABC$.

2007 All-Russian Olympiad Regional Round, 8.7

Tags: geometry , circumcircle , incenter , geometry proposed

Given an isosceles triangle $ ABC$ with $ AB \equal{} BC$. A point $ M$ is chosen inside $ ABC$ such that $ \angle AMC \equal{} 2\angle ABC$ . A point $ K$ lies on segment $ AM$ such that $ \angle BKM \equal{}\angle ABC$. Prove that $ BK \equal{} KM\plus{}MC$.

2005 Taiwan TST Round 3, 2

Tags: geometry , parallelogram , rhombus , geometry proposed

It is known that $\triangle ABC$ is an acute triangle. Let $C'$ be the foott of the perpendicular from $C$ to $AB$, and $D$, $E$ two distinct points on $CC'$. The feet of the perpendiculars from $D$ to $AC$ and $BC$ are $F$ and $G$, respectively. Show that if $DGEF$ is a parallelogram then $ABC$ is isosceles.

1996 Baltic Way, 5

Tags: geometry , incenter , rectangle , Euler , geometry proposed

Let $ABCD$ be a cyclic convex quadrilateral and let $r_a,r_b,r_c,r_d$ be the radii of the circles inscribed in the triangles $BCD, ACD, ABD, ABC$, respectively. Prove that $r_a+r_c=r_b+r_d$.

2013 Hitotsubashi University Entrance Examination, 4

Tags: quadratics , geometry proposed , geometry

Let $t$ be a positive constant. Given two points $A(2t,\ 2t,\ 0),\ B(0,\ 0,\ t)$ in a space with the origin $O$. Suppose mobile points $P$ in such way that $\overrightarrow{OP}\cdot \overrightarrow{AP}+\overrightarrow{OP}\cdot \overrightarrow{BP}+\overrightarrow{AP}\cdot \overrightarrow{BP}=3.$ Find the value of $t$ such that the maximum value of $OP$ is 3.

2008 China Team Selection Test, 1

Tags: geometry , rectangle , geometry proposed

Given a rectangle $ ABCD,$ let $ AB \equal{} b, AD \equal{} a ( a\geq b),$ three points $ X,Y,Z$ are put inside or on the boundary of the rectangle, arbitrarily. Find the maximum of the minimum of the distances between any two points among the three points. (Denote it by $ a,b$)