Olympiad problems

2010 Kurschak Competition, 2

Consider a triangle $ABC$, with the points $A_1$, $A_2$ on side $BC$, $B_1,B_2\in\overline{AC}$, $C_1,C_2\in\overline{AB}$ such that $AC_1<AC_2$, $BA_1<BA_2$, $CB_1<CB_2$. Let the circles $AB_1C_1$ and $AB_2C_2$ meet at $A$ and $A^*$. Similarly, let the circles $BC_1A_1$ and $BC_2A_2$ intersect at $B^*\neq B$, let $CA_1B_1$ and $CA_2B_2$ intersect at $C^*\neq C$. Prove that the lines $AA^*$, $BB^*$, $CC^*$ are concurrent.

1998 China Team Selection Test, 1

Tags: geometry , circumcircle , incenter , trigonometry , geometry unsolved

In acute-angled $\bigtriangleup ABC$, $H$ is the orthocenter, $O$ is the circumcenter and $I$ is the incenter. Given that $\angle C > \angle B > \angle A$, prove that $I$ lies within $\bigtriangleup BOH$.

2014 Contests, 2

Tags: geometry , circumcircle , incenter , cyclic quadrilateral , geometry unsolved

Let $ABCD$ be a convex cyclic quadrilateral with $AD=BD$. The diagonals $AC$ and $BD$ intersect in $E$. Let the incenter of triangle $\triangle BCE$ be $I$. The circumcircle of triangle $\triangle BIE$ intersects side $AE$ in $N$. Prove \[ AN \cdot NC = CD \cdot BN. \]

2010 Germany Team Selection Test, 1

Tags: geometry unsolved , geometry

In the plane we have points $P,Q,A,B,C$ such triangles $APQ,QBP$ and $PQC$ are similar accordantly (same direction). Then let $A'$ ($B',C'$ respectively) be the intersection of lines $BP$ and $CQ$ ($CP$ and $AQ;$ $AP$ and $BQ,$ respectively.) Show that the points $A,B,C,A',B',C'$ lie on a circle.

2012 Sharygin Geometry Olympiad, 9

Tags: geometry , geometric transformation , reflection , geometry unsolved

In triangle $ABC$, given lines $l_{b}$ and $l_{c}$ containing the bisectors of angles $B$ and $C$, and the foot $L_{1}$ of the bisector of angle $A$. Restore triangle $ABC$.

2007 India National Olympiad, 5

Tags: inequalities , geometry unsolved , geometry

Let $ ABC$ be a triangle in which $ AB\equal{}AC$. Let $ D$ be the midpoint of $ BC$ and $ P$ be a point on $ AD$. Suppose $ E$ is the foot of perpendicular from $ P$ on $ AC$. Define \[ \frac{AP}{PD}\equal{}\frac{BP}{PE}\equal{}\lambda , \ \ \ \frac{BD}{AD}\equal{}m , \ \ \ z\equal{}m^2(1\plus{}\lambda)\] Prove that \[ z^2 \minus{} (\lambda^3 \minus{} \lambda^2 \minus{} 2)z \plus{} 1 \equal{} 0\] Hence show that $ \lambda \ge 2$ and $ \lambda \equal{} 2$ if and only if $ ABC$ is equilateral.

1998 Iran MO (3rd Round), 3

Tags: analytic geometry , geometry unsolved , geometry

Let $n(r)$ be the maximum possible number of points with integer coordinates on a circle with radius $r$ in Cartesian plane. Prove that $n(r) < 6\sqrt[3]{3 \pi r^2}.$

2002 China Team Selection Test, 1

Tags: inequalities , trigonometry , geometry unsolved , geometry

In acute triangle $ ABC$, show that: $ \sin^3{A}\cos^2{(B \minus{} C)} \plus{} \sin^3{B}\cos^2{(C \minus{} A)} \plus{} \sin^3{C}\cos^2{(A \minus{} B)} \leq 3\sin{A} \sin{B} \sin{C}$ and find out when the equality holds.

1984 IMO Longlists, 62

Tags: geometry , circumcircle , geometry unsolved

From a point $P$ exterior to a circle $K$, two rays are drawn intersecting $K$ in the respective pairs of points $A, A'$ and $B,B' $. For any other pair of points $C, C'$ on $K$, let $D$ be the point of intersection of the circumcircles of triangles $PAC$ and $PB'C'$ other than point $P$. Similarly, let $D'$ be the point of intersection of the circumcircles of triangles $PA'C'$ and $PBC$ other than point $P$. Prove that the points $P, D$, and $D'$ are collinear.

2003 Brazil National Olympiad, 1

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

Given a circle and a point $A$ inside the circle, but not at its center. Find points $B$, $C$, $D$ on the circle which maximise the area of the quadrilateral $ABCD$.

1997 Romania Team Selection Test, 2

Tags: function , geometry unsolved , geometry

Let $P$ be the set of points in the plane and $D$ the set of lines in the plane. Determine whether there exists a bijective function $f: P \rightarrow D$ such that for any three collinear points $A$, $B$, $C$, the lines $f(A)$, $f(B)$, $f(C)$ are either parallel or concurrent. [i]Gefry Barad[/i]

2010 Czech-Polish-Slovak Match, 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 $80$.

2004 Rioplatense Mathematical Olympiad, Level 3, 3

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

In a convex hexagon $ABCDEF$, triangles $ACE$ and $BDF$ have the same circumradius $R$. If triangle $ACE$ has inradius $r$, prove that \[ \text{Area}(ABCDEF)\le\frac{R}{r}\cdot\text{Area}(ACE).\]

1992 Hungary-Israel Binational, 4

Tags: analytic geometry , geometry , parallelogram , geometry unsolved

We are given a convex pentagon $ABCDE$ in the coordinate plane such that $A$, $B$, $C$, $D$, $E$ are lattice points. Let $Q$ denote the convex pentagon bounded by the five diagonals of the pentagon $ABCDE$ (so that the vertices of $Q$ are the interior points of intersection of diagonals of the pentagon $ABCDE$). Prove that there exists a lattice point inside of $Q$ or on the boundary of $Q$.

1994 Turkey MO (2nd round), 6

Tags: geometry , incenter , perimeter , trigonometry , trapezoid , angle bisector , geometry unsolved

The incircle of triangle $ABC$ touches $BC$ at $D$ and $AC$ at $E$. Let $K$ be the point on $CB$ with $CK=BD$, and $L$ be the point on $CA$ with $AE=CL$. Lines $AK$ and $BL$ meet at $P$. If $Q$ is the midpoint of $BC$, $I$ the incenter, and $G$ the centroid of $\triangle ABC$, show that: $(a)$ $IQ$ and $AK$ are parallel, $(b)$ the triangles $AIG$ and $QPG$ have equal area.

2014 Sharygin Geometry Olympiad, 6

Tags: geometry , perpendicular bisector , geometry unsolved

Given a circle with center $O$ and a point $P$ not lying on it, let $X$ be an arbitrary point on this circle and $Y$ be a common point of the bisector of angle $POX$ and the perpendicular bisector to segment $PX$. Find the locus of points $Y$.

2010 Lithuania National Olympiad, 2

Tags: geometry , trapezoid , angle bisector , geometry unsolved

In trapezoid $ABCD$, $AD$ is parallel to $BC$. Knowing that $AB=AD+BC$, prove that the bisector of $\angle A$ also bisects $CD$.

1984 IMO Longlists, 2

Tags: modular arithmetic , geometry unsolved , geometry

Given a regular convex $2m$- sided polygon $P$, show that there is a $2m$-sided polygon $\pi$ with the same vertices as $P$ (but in different order) such that $\pi$ has exactly one pair of parallel sides.

2001 Tournament Of Towns, 2

Tags: geometry , perimeter , geometry unsolved

Let $n\ge3$ be an integer. A circle is divided into $2n$ arcs by $2n$ points. Each arc has one of three possible lengths, and no two adjacent arcs have the same lengths. The $2n$ points are colored alternately red and blue. Prove that the $n$-gon with red vertices and the $n$-gon with blue vertices have the same perimeter and the same area.

2012 Korea - Final Round, 2

Tags: geometry , incenter , symmetry , analytic geometry , projective geometry , geometry unsolved

For a triangle $ ABC $ which $ \angle B \ne 90^{\circ} $ and $ AB \ne AC $, define $ P_{ABC} $ as follows ; Let $ I $ be the incenter of triangle $ABC$, and let $ D, E, F $ be the intersection points with the incircle and segments $ BC, CA, AB $. Two lines $ AB $ and $ DI $ meet at $ S $ and let $ T $ be the intersection point of line $ DE $ and the line which is perpendicular with $ DF $ at $ F $. The line $ ST $ intersects line $ EF $ at $ R$. Now define $ P_{ABC} $ be one of the intersection points of the incircle and the circle with diameter $ IR $, which is located in other side with $ A $ about $ IR $. Now think of an isosceles triangle $ XYZ $ such that $ XZ = YZ > XY $. Let $ W $ be the point on the side $ YZ $ such that $ WY < XY $ and Let $ K = P_{YXW} $ and $ L = P_{ZXW} $. Prove that $ 2 KL \le XY $.

2005 Korea National Olympiad, 2

Tags: geometry , circumcircle , geometry unsolved

For triangle $ABC$, $P$ and $Q$ satisfy $\angle BPA + \angle AQC=90^{\circ}$. It is provided that the vertices of the triangle $BAP$ and $ACQ$ are ordered counterclockwise(or clockwise). Let the intersection of the circumcircles of the two triangles be $N$ ($A \neq N$, however if $A$ is the only intersection $A=N$), and the midpoint of segment $BC$ be $M$. Show that the length of $MN$ does not depend on $P$ and $Q$.

1971 IMO Longlists, 9

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

The base of an inclined prism is a triangle $ABC$. The perpendicular projection of $B_1$, one of the top vertices, is the midpoint of $BC$. The dihedral angle between the lateral faces through $BC$ and $AB$ is $\alpha$, and the lateral edges of the prism make an angle $\beta$ with the base. If $r_1, r_2, r_3$ are exradii of a perpendicular section of the prism, assuming that in $ABC, \cos^2 A + \cos^2 B + \cos^2 C = 1, \angle A < \angle B < \angle C,$ and $BC = a$, calculate $r_1r_2 + r_1r_3 + r_2r_3.$

1988 IMO Longlists, 71

Tags: trigonometry , geometry unsolved , geometry

The quadrilateral $A_1A_2A_3A_4$ is cyclic, and its sides are $a_1 = A_1A_2, a_2 = A_2A_3, a_3 = A_3A_4$ and $a_4 = A_4A_1.$ The respective circles with centres $I_i$ and radii $r_i$ are tangent externally to each side $a_i$ and to the sides $a_{i+1}$ and $a_{i-1}$ extended. ($a_0 = a_4$). Show that \[ \prod^4_{i=1} \frac{a_i}{r_i} = 4 \cdot (\csc (A_1) + \csc (A_2) )^2. \]

2006 Peru IMO TST, 4

Tags: geometry , circumcircle , LaTeX , parallelogram , perpendicular bisector , similar triangles , geometry unsolved

[color=blue][size=150]PERU TST IMO - 2006[/size] Saturday, may 20.[/color] [b]Question 04[/b] In an actue-angled triangle $ABC$ draws up: its circumcircle $w$ with center $O$, the circumcircle $w_1$ of the triangle $AOC$ and the diameter $OQ$ of $w_1$. The points are chosen $M$ and $N$ on the straight lines $AQ$ and $AC$, respectively, in such a way that the quadrilateral $AMBN$ is a parallelogram. Prove that the intersection point of the straight lines $MN$ and $BQ$ belongs the circumference $w_1.$ --- [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=88513]Spanish version[/url] $\text{\LaTeX}{}$ed by carlosbr

1994 APMO, 2

Tags: geometry , circumcircle , Euler , analytic geometry , perpendicular bisector , complex numbers , geometry unsolved

Given a nondegenerate triangle $ABC$, with circumcentre $O$, orthocentre $H$, and circumradius $R$, prove that $|OH| < 3R$.