Olympiad problems

2004 China Team Selection Test, 1

Find the largest value of the real number $ \lambda$, such that as long as point $ P$ lies in the acute triangle $ ABC$ satisfying $ \angle{PAB}\equal{}\angle{PBC}\equal{}\angle{PCA}$, and rays $ AP$, $ BP$, $ CP$ intersect the circumcircle of triangles $ PBC$, $ PCA$, $ PAB$ at points $ A_1$, $ B_1$, $ C_1$ respectively, then $ S_{A_1BC}\plus{} S_{B_1CA}\plus{} S_{C_1AB} \geq \lambda S_{ABC}$.

2010 BMO TST, 3

Tags: geometry , trapezoid , perimeter , circumcircle , geometry unsolved

Let $ K$ be the circumscribed circle of the trapezoid $ ABCD$ . In this trapezoid the diagonals $ AC$ and $ BD$ are perpendicular. The parallel sides $ AB\equal{}a$ and $ CD\equal{}c$ are diameters of the circles $ K_{a}$ and $ K_{b}$ respectively. Find the perimeter and the area of the part inside the circle $ K$, that is outside circles $ K_{a}$ and $ K_{b}$.

2010 Indonesia TST, 3

Tags: geometry , geometry unsolved

Given acute triangle $ABC$ with circumcenter $O$ and the center of nine-point circle $N$. Point $N_1$ are given such that $\angle NAB = \angle N_1AC$ and $\angle NBC = \angle N_1BA$. Perpendicular bisector of segment $OA$ intersects the line $BC$ at $A_1$. Analogously define $B_1$ and $C_1$. Show that all three points $A_1,B_1,C_1$ are collinear at a line that is perpendicular to $ON_1$.

1983 IMO Longlists, 65

Tags: trigonometry , geometry unsolved , geometry

Let $ABCD$ be a convex quadrilateral whose diagonals $AC$ and $BD$ intersect in a point $P$. Prove that \[\frac{AP}{PC}=\frac{\cot \angle BAC + \cot \angle DAC}{\cot \angle BCA + \cot \angle DCA}\]

2011 Indonesia MO, 8

Tags: geometry , circumcircle , geometry unsolved

Given a triangle $ABC$. Its incircle is tangent to $BC, CA, AB$ at $D, E, F$ respectively. Let $K, L$ be points on $CA, AB$ respectively such that $K \neq A \neq L, \angle EDK = \angle ADE, \angle FDL = \angle ADF$. Prove that the circumcircle of $AKL$ is tangent to the incircle of $ABC$.

1985 Kurschak Competition, 3

Tags: geometry , geometric transformation , reflection , circumcircle , homothety , inequalities , geometry unsolved

We reflected each vertex of a triangle on the opposite side. Prove that the area of the triangle formed by these three reflection points is smaller than the area of the initial triangle multiplied by five.

1995 Polish MO Finals, 2

Tags: geometry , geometry unsolved

The diagonals of a convex pentagon divide it into a small pentagon and ten triangles. What is the largest number of the triangles that can have the same area?

2003 Czech-Polish-Slovak Match, 4

Tags: geometry unsolved , geometry

Point $P$ lies on the median from vertex $C$ of a triangle $ABC$. Line $AP$ meets $BC$ at $X$, and line $BP$ meets $AC$ at $Y$ . Prove that if quadrilateral $ABXY$ is cyclic, then triangle $ABC$ is isosceles.

2008 China National Olympiad, 1

Tags: geometry , circumcircle , incenter , geometric transformation , reflection , perpendicular bisector , geometry unsolved

Suppose $\triangle ABC$ is scalene. $O$ is the circumcenter and $A'$ is a point on the extension of segment $AO$ such that $\angle BA'A = \angle CA'A$. Let point $A_1$ and $A_2$ be foot of perpendicular from $A'$ onto $AB$ and $AC$. $H_{A}$ is the foot of perpendicular from $A$ onto $BC$. Denote $R_{A}$ to be the radius of circumcircle of $\triangle H_{A}A_1A_2$. Similiarly we can define $R_{B}$ and $R_{C}$. Show that: \[\frac{1}{R_{A}} + \frac{1}{R_{B}} + \frac{1}{R_{C}} = \frac{2}{R}\] where R is the radius of circumcircle of $\triangle ABC$.

2009 Germany Team Selection Test, 2

Tags: geometry , circumcircle , incenter , geometry unsolved

Let triangle $ABC$ be perpendicular at $A.$ Let $M$ be the midpoint of segment $\overline{BC}.$ Point $D$ lies on side $\overline{AC}$ and satisfies $|AD|=|AM|.$ Let $P \neq C$ be the intersection of the circumcircle of triangles $AMC$ and $BDC.$ Prove that $CP$ bisects the angle at $C$ of triangle $ABC.$

1997 India National Olympiad, 4

Tags: geometry , perimeter , inequalities , triangle inequality , geometry unsolved

In a unit square one hundred segments are drawn from the centre to the sides dividing the square into one hundred parts (triangles and possibly quadruilaterals). If all parts have equal perimetr $p$, show that $\dfrac{14}{10} < p < \dfrac{15}{10}$.

1979 IMO Longlists, 11

Tags: geometry , 3D geometry , pyramid , geometry unsolved

Prove that a pyramid $A_1A_2 \ldots A_{2k+1}S$ with equal lateral edges and equal space angles between adjacent lateral walls is regular.

2010 Contests, 3

Tags: geometry , rectangle , exterior angle , geometry unsolved

An angle is given in a plane. Using only a compass, one must find out $(a)$ if this angle is acute. Find the minimal number of circles one must draw to be sure. $(b)$ if this angle equals $31^{\circ}$.(One may draw as many circles as one needs).

2010 Tournament Of Towns, 6

Tags: geometry unsolved , geometry

A broken line consists of $31$ segments. It has no self intersections, and its start and end points are distinct. All segments are extended to become straight lines. Find the least possible number of straight lines.

2004 China Team Selection Test, 2

Tags: geometry , perimeter , inequalities , trigonometry , trig identities , Law of Sines , geometry unsolved

Convex quadrilateral $ ABCD$ is inscribed in a circle, $ \angle{A}\equal{}60^o$, $ BC\equal{}CD\equal{}1$, rays $ AB$ and $ DC$ intersect at point $ E$, rays $ BC$ and $ AD$ intersect each other at point $ F$. It is given that the perimeters of triangle $ BCE$ and triangle $ CDF$ are both integers. Find the perimeter of quadrilateral $ ABCD$.

2008 Nordic, 3

Tags: geometry , circumcircle , geometry unsolved

Let $ABC$ be a triangle and $D,E$ be points on $BC,CA$ such that $AD,BE$ are angle bisectors of $\triangle ABC$. Let $F,G$ be points on the circumcircle of $\triangle ABC$ such that $AF||DE$ and $FG||BC$. Prove that $\frac{AG}{BG}= \frac{AB+AC}{AB+BC}$.

2013 ELMO Shortlist, 6

Tags: trigonometry , projective geometry , geometry unsolved , geometry

Let $ABCDEF$ be a non-degenerate cyclic hexagon with no two opposite sides parallel, and define $X=AB\cap DE$, $Y=BC\cap EF$, and $Z=CD\cap FA$. Prove that \[\frac{XY}{XZ}=\frac{BE}{AD}\frac{\sin |\angle{B}-\angle{E}|}{\sin |\angle{A}-\angle{D}|}.\][i]Proposed by Victor Wang[/i]

2000 Kurschak Competition, 2

Tags: geometry , circumcircle , geometry unsolved

Let $ABC$ be a non-equilateral triangle in the plane, and let $T$ be a point different from its vertices. Define $A_T$, $B_T$ and $C_T$ as the points where lines $AT$, $BT$, and $CT$ meet the circumcircle of $ABC$. Prove that there are exactly two points $P$ and $Q$ in the plane for which the triangles $A_PB_PC_P$ and $A_QB_QC_Q$ are equilateral. Prove furthermore that line $PQ$ contains the circumcenter of $\triangle ABC$.

1998 Irish Math Olympiad, 4

Tags: geometry unsolved , geometry

Show that a disk of radius $ 2$ can be covered by seven (possibly overlapping) disks of radius $ 1$.

2005 MOP Homework, 6

Tags: geometry , incenter , geometry unsolved

Consider the three disjoint arcs of a circle determined by three points of the circle. We construct a circle around each of the midpoint of every arc which goes the end points of the arc. Prove that the three circles pass through a common point.

2008 Turkey MO (2nd round), 2

Tags: geometry , symmetry , geometry unsolved

A circle $ \Gamma$ and a line $ \ell$ is given in a plane such that $ \ell$ doesn't cut $ \Gamma$.Determine the intersection set of the circles has $ [AB]$ as diameter for all pairs of $ \left\{A,B\right\}$ (lie on $ \ell$) and satisfy $ P,Q,R,S \in \Gamma$ such that $ PQ \cap RS\equal{}\left\{A\right\}$ and $ PS \cap QR\equal{}\left\{B\right\}$

1999 Finnish National High School Mathematics Competition, 4

Tags: geometry , circumcircle , ratio , geometry unsolved

Three unit circles have a common point $O.$ The other points of (pairwise) intersection are $A, B$ and $C$. Show that the points $A, B$ and $C$ are located on some unit circle.

2005 International Zhautykov Olympiad, 2

Tags: geometry unsolved , geometry

The inner point $ X$ of a quadrilateral is [i]observable[/i] from the side $ YZ$ if the perpendicular to the line $ YZ$ meet it in the colosed interval $ [YZ].$ The inner point of a quadrilateral is a $ k\minus{}$point if it is observable from the exactly $ k$ sides of the quadrilateral. Prove that if a convex quadrilateral has a 1-point then it has a $ k\minus{}$point for each $ k\equal{}2,3,4.$

1994 Vietnam National Olympiad, 2

Tags: geometry , 3D geometry , sphere , tetrahedron , vector , analytic geometry , geometry unsolved

$S$ is a sphere center $O. G$ and $G'$ are two perpendicular great circles on $S$. Take $A, B, C$ on $G$ and $D$ on $G'$ such that the altitudes of the tetrahedron $ABCD$ intersect at a point. Find the locus of the intersection.

1995 Kurschak Competition, 1

Tags: geometry , rectangle , analytic geometry , geometry unsolved

Given in the plane is a lattice and a grid rectangle with sides parallel to the coordinate axes. We divide the rectangle into grid triangles with area $\frac12$. Prove that the number of right angled triangles is at least twice as much as the shorter side of the rectangle. (A grid polygon is a polygon such that both coordinates of each vertex is an integer.)