Olympiad problems

2006 India IMO Training Camp, 3

Let $ABC$ be an equilateral triangle, and let $D,E$ and $F$ be points on $BC,BA$ and $AB$ respectively. Let $\angle BAD= \alpha, \angle CBE=\beta$ and $\angle ACF =\gamma$. Prove that if $\alpha+\beta+\gamma \geq 120^\circ$, then the union of the triangular regions $BAD,CBE,ACF$ covers the triangle $ABC$.

2008 Peru IMO TST, 1

Tags: geometry , geometric transformation , reflection , geometry unsolved

Let $ ABC$ be a triangle and let $ I$ be the incenter. $ Ia$ $ Ib$ and $ Ic$ are the excenters opposite to points $ A$ $ B$ and $ C$ respectively. Let $ La$ be the line joining the orthocenters of triangles $ IBC$ and $ IaBC$. Define $ Lb$ and $ Lc$ in the same way. Prove that $ La$ $ Lb$ and $ Lc$ are concurrent. Daniel

1975 Vietnam National Olympiad, 6

Tags: geometry unsolved , geometry

Let us have a line $\ell$ in the space and a point $A$ not lying on $\ell.$ For an arbitrary line $\ell'$ passing through $A$, $XY$ ($Y$ is on $\ell'$) is a common perpendicular to the lines $\ell$ and $\ell'.$ Find the locus of points $Y.$

2011 India IMO Training Camp, 1

Tags: geometry , trapezoid , trigonometry , cyclic quadrilateral , geometry unsolved

Let $ABC$ be a triangle each of whose angles is greater than $30^{\circ}$. Suppose a circle centered with $P$ cuts segments $BC$ in $T,Q; CA$ in $K,L$ and $AB$ in $M,N$ such that they are on a circle in counterclockwise direction in that order.Suppose further $PQK,PLM,PNT$ are equilateral. Prove that: $a)$ The radius of the circle is $\frac{2abc}{a^2+b^2+c^2+4\sqrt{3}S}$ where $S$ is area. $b) a\cdot AP=b\cdot BP=c\cdot PC.$

1993 China National Olympiad, 3

Tags: geometry unsolved , geometry

Let $K, K_1$ be two circles with the same center and their radii equal to $R$ and $R_1 (R_1>R)$ respectively. Quadrilateral $ABCD$ is inscribed in circle $K$. Quadrilateral $A_1B_1C_1D_1$ is inscribed in circle $K_1$ where $A_1,B_1,C_1,D_1$ lie on rays $CD,DA,AB,BC$ respectively. Show that $\dfrac{S_{A_1B_1C_1D_1}}{S_{ABCD}}\ge \dfrac{R^2_1}{R^2}$.

2003 Moldova Team Selection Test, 3

Tags: geometry , geometry unsolved

Consider a point $ M$ found in the same plane with the triangle $ ABC$, but not found on any of the lines $ AB,BC$ and $ CA$. Denote by $ S_1,S_2$ and $ S_3$ the areas of the triangles $ AMB,BMC$ and $ CMA$, respectively. Find the locus of $ M$ satisfying the relation: $ (MA^2\plus{}MB^2\plus{}MC^2)^2\equal{}16(S_1^2\plus{}S_2^2\plus{}S_3^2)$

Indonesia Regional MO OSP SMA - geometry, 2008.3

Tags: geometry , trigonometry , geometry unsolved

Given triangle $ ABC$. The incircle of triangle $ ABC$ is tangent to $ BC,CA,AB$ at $ D,E,F$ respectively. Construct point $ G$ on $ EF$ such that $ DG$ is perpendicular to $ EF$. Prove that $ \frac{FG}{EG} \equal{} \frac{BF}{CE}$.

1971 IMO Longlists, 25

Tags: analytic geometry , geometry unsolved , geometry

Let $ABC,AA_1A_2,BB_1B_2, CC_1C_2$ be four equilateral triangles in the plane satisfying only that they are all positively oriented (i.e., in the counterclockwise direction). Denote the midpoints of the segments $A_2B_1,B_2C_1, C_2A_1$ by $P,Q,R$ in this order. Prove that the triangle $PQR$ is equilateral.

2005 MOP Homework, 1

Tags: geometry unsolved , geometry

Isosceles triangle $ABC$, with $AB=AC$, is inscribed in circle $\omega$. Point $D$ lies on arc $\frown{BC}$ not containing $A$. Let $E$ be the foot of perpendicular from $A$ to line $CD$. Prove that $BC+DC=2DE$.

2008 All-Russian Olympiad, 6

Tags: geometry , circumcircle , trigonometry , search , geometry unsolved

The incircle of a triangle $ABC$ touches the side $AB$ and $AC$ at respectively at $X$ and $Y$. Let $K$ be the midpoint of the arc $\widehat{AB}$ on the circumcircle of $ABC$. Assume that $XY$ bisects the segment $AK$. What are the possible measures of angle $BAC$?

2006 Nordic, 1

Tags: geometry , circumcircle , geometric transformation , perpendicular bisector , angle bisector , geometry unsolved

Points $B,C$ vary on two fixed rays emanating from point $A$ such that $AB+AC$ is constant. Show that there is a point $D$, other than $A$, such that the circumcircle of triangle $ABC$ passes through $D$ for all possible choices of $B, C$.

1996 Vietnam Team Selection Test, 1

Tags: symmetry , geometry unsolved , geometry

Given 3 non-collinear points $A,B,C$. For each point $M$ in the plane ($ABC$) let $M_1$ be the point symmetric to $M$ with respect to $AB$, $M_2$ be the point symmetric to $M_1$ with respect to $BC$ and $M'$ be the point symmetric to $M_2$ with respect to $AC$. Find all points $M$ such that $MM'$ obtains its minimum. Let this minimum value be $d$. Prove that $d$ does not depend on the order of the axes of symmetry we chose (we have 3 available axes, that is $BC$, $CA$, $AB$. In the first part the order of axes we chose $AB$, $BC$, $CA$, and the second part of the problem states that the value $d$ doesn't depend on this order).

2010 Contests, 2

Tags: geometry unsolved , geometry

Given any $60$ points on a circle of radius $1$, prove that there is a point on the circle the sum of whose distances to these $60$ points is at most $80$.

2012 Kazakhstan National Olympiad, 2

Tags: geometry , circumcircle , geometry unsolved

Given two circles $k_{1}$ and $k_{2}$ with centers $O_{1}$ and $O_{2}$ that intersect at the points $A$ and $B$.Passes through A two lines that intersect the circle $k_{1}$ at the points $N_{1}$and $M_{1}$, and the circle $k_{2}$ at the points $N_{2}$ and $M_{2}$ (points $A, N_{1},M_{1}$ in colinear). Denote the midpoints of the segments $N_{1}N_{2}$ and $M_{1}M_{2]}$ , through $N$ and $M$.Prove that: $a)$ Points $M,N,A$ and $B$ lie on a circle $b)$The center of the circle passing through $M,N,A$ and $B$ lies in the middle of the segment $O_{1}O_{2}$

2010 Sharygin Geometry Olympiad, 19

Tags: geometry , parallelogram , circumcircle , geometry unsolved

A quadrilateral $ABCD$ is inscribed into a circle with center $O.$ Points $P$ and $Q$ are opposite to $C$ and $D$ respectively. Two tangents drawn to that circle at these points meet the line $AB$ in points $E$ and $F.$ ($A$ is between $E$ and $B$, $B$ is between $A$ and $F$). The line $EO$ meets $AC$ and $BC$ in points $X$ and $Y$ respectively, and the line $FO$ meets $AD$ and $BD$ in points $U$ and $V$ respectively. Prove that $XV=YU.$

2013 ELMO Shortlist, 14

Tags: function , geometry , perimeter , geometry unsolved

Let $O$ be a point (in the plane) and $T$ be an infinite set of points such that $|P_1P_2| \le 2012$ for every two distinct points $P_1,P_2\in T$. Let $S(T)$ be the set of points $Q$ in the plane satisfying $|QP| \le 2013$ for at least one point $P\in T$. Now let $L$ be the set of lines containing exactly one point of $S(T)$. Call a line $\ell_0$ passing through $O$ [i]bad[/i] if there does not exist a line $\ell\in L$ parallel to (or coinciding with) $\ell_0$. (a) Prove that $L$ is nonempty. (b) Prove that one can assign a line $\ell(i)$ to each positive integer $i$ so that for every bad line $\ell_0$ passing through $O$, there exists a positive integer $n$ with $\ell(n) = \ell_0$. [i]Proposed by David Yang[/i]

2000 Baltic Way, 3

Tags: geometry , circumcircle , trigonometry , geometry unsolved

Given a triangle $ ABC$ with $ \angle A \equal{} 90^{\circ}$ and $ AB \neq AC$. The points $ D$, $ E$, $ F$ lie on the sides $ BC$, $ CA$, $ AB$, respectively, in such a way that $ AFDE$ is a square. Prove that the line $ BC$, the line $ FE$ and the line tangent at the point $ A$ to the circumcircle of the triangle $ ABC$ intersect in one point.

2002 Brazil National Olympiad, 5

Tags: geometry , geometry unsolved

A finite collection of squares has total area $4$. Show that they can be arranged to cover a square of side $1$.

2014 Tuymaada Olympiad, 3

Tags: geometry , geometric transformation , homothety , reflection , geometry unsolved , Tuymaada

The points $K$ and $L$ on the side $BC$ of a triangle $\triangle{ABC}$ are such that $\widehat{BAK}=\widehat{CAL}=90^\circ$. Prove that the midpoint of the altitude drawn from $A$, the midpoint of $KL$ and the circumcentre of $\triangle{ABC}$ are collinear. [i](A. Akopyan, S. Boev, P. Kozhevnikov)[/i]

1987 China National Olympiad, 4

Tags: geometry , geometry unsolved

Five points are arbitrarily put inside a given equilateral triangle $ABC$ whose area is equal to $1$. Show that we can draw three equilateral triangles within triangle $ABC$ such that the following conditions are all satisfied: i) the five points are covered by the three equilateral triangles; ii) any side of the three equilateral triangles is parallel to a certain side of the triangle $ABC$; iii) the sum of the areas of the three equilateral triangles is not larger than $0.64$.

2002 China Team Selection Test, 1

Tags: geometry , trapezoid , geometry unsolved

Circle $ O$ is inscribed in a trapzoid $ ABCD$, $ \angle{A}$ and $ \angle{B}$ are all acute angles. A line through $ O$ intersects $ AD$ at $ E$ and $ BC$ at $ F$, and satisfies the following conditions: (1) $ \angle{DEF}$ and $ \angle{CFE}$ are acute angles. (2) $ AE\plus{}BF\equal{}DE\plus{}CF$. Let $ AB\equal{}a$, $ BC\equal{}b$, $ CD\equal{}c$, then use $ a,b,c$ to express $ AE$.

2009 Iran MO (3rd Round), 7

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

A sphere is inscribed in polyhedral $P$. The faces of $P$ are coloured with black and white in a way that no two black faces share an edge. Prove that the sum of surface of black faces is less than or equal to the sum of the surface of the white faces. Time allowed for this problem was 1 hour.

1981 Bundeswettbewerb Mathematik, 2

Tags: geometry unsolved , geometry

A [b] bijective[/b] mapping from a plane to itself maps every circle to a circle. Prove that it maps every line to a line.

2006 IberoAmerican, 2

Tags: geometry , cyclic quadrilateral , projective geometry , geometry unsolved

[color=darkred]The sides $AD$ and $CD$ of a tangent quadrilateral $ABCD$ touch the incircle $\varphi$ at $P$ and $Q,$ respectively. If $M$ is the midpoint of the chord $XY$ determined by $\varphi$ on the diagonal $BD,$ prove that $\angle AMP = \angle CMQ.$[/color]

2000 Cono Sur Olympiad, 2

Tags: geometry , geometric transformation , rotation , analytic geometry , invariant , geometry unsolved

Consider the following transformation of the Cartesian plane: choose a lattice point and rotate the plane $90^\circ$ counterclockwise about that lattice point. Is it possible, through a sequence of such transformations, to take the triangle with vertices $(0,0)$, $(1,0)$ and $(0,1)$ to the triangle with vertices $(0,0)$, $(1,0)$ and $(1,1)$?