Olympiad problems

2011 Mexico National Olympiad, 2

Tags: geometry , circumcircle , geometric transformation , reflection , power of a point , perpendicular bisector , geometry proposed

Let $ABC$ be an acute triangle and $\Gamma$ its circumcircle. Let $l$ be the line tangent to $\Gamma$ at $A$. Let $D$ and $E$ be the intersections of the circumference with center $B$ and radius $AB$ with lines $l$ and $AC$, respectively. Prove the orthocenter of $ABC$ lies on line $DE$.

2005 Taiwan TST Round 2, 3

Tags: conics , ellipse , geometry proposed , geometry

In the interior of an ellipse with major axis 2 and minor axis 1, there are more than 6 segments with total length larger than 15. Prove that there exists a line passing through all of the segments.

2001 Baltic Way, 6

Tags: geometry , parallelogram , geometry proposed

The points $A, B, C, D, E$ lie on the circle $c$ in this order and satisfy $AB\parallel EC$ and $AC\parallel ED$. The line tangent to the circle $c$ at $E$ meets the line $AB$ at $P$. The lines $BD$ and $EC$ meet at $Q$. Prove that $|AC|=|PQ|$.

1967 IMO Longlists, 4

Tags: inequalities , trigonometry , geometry proposed , geometry

Suppose, medians $m_a$ and $m_b$ of a triangle are orthogonal. Prove that: (a) The medians of the triangle correspond to the sides of a right-angled triangle. (b) If $a,b,c$ are the side-lengths of the triangle, then, the following inequality holds:\[5(a^2+b^2-c^2)\geq 8ab\]

2002 Tournament Of Towns, 5

Tags: projective geometry , geometry , cyclic quadrilateral , geometry proposed

Two circles $\Gamma_1,\Gamma_2$ intersect at $A,B$. Through $B$ a straight line $\ell$ is drawn and $\ell\cap \Gamma_1=K,\ell\cap\Gamma_2=M\;(K,M\neq B)$. We are given $\ell_1\parallel AM$ is tangent to $\Gamma_1$ at $Q$. $QA\cap \Gamma_2=R\;(\neq A)$ and further $\ell_2$ is tangent to $\Gamma_2$ at $R$. Prove that: [list] [*]$\ell_2\parallel AK$ [*]$\ell,\ell_1,\ell_2$ have a common point.[/list]

2013 Tuymaada Olympiad, 8

Tags: inequalities , geometry , perimeter , analytic geometry , trigonometry , geometry proposed

The point $A_1$ on the perimeter of a convex quadrilateral $ABCD$ is such that the line $AA_1$ divides the quadrilateral into two parts of equal area. The points $B_1$, $C_1$, $D_1$ are defined similarly. Prove that the area of the quadrilateral $A_1B_1C_1D_1$ is greater than a quarter of the area of $ABCD$. [i]L. Emelyanov [/i]

VMEO IV 2015, 10.2

Tags: circumcircle , geometry proposed , geometry

Given triangle $ABC$ and $P,Q$ are two isogonal conjugate points in $\triangle ABC$. $AP,AQ$ intersects $(QBC)$ and $(PBC)$ at $M,N$, respectively ( $M,N$ be inside triangle $ABC$) 1. Prove that $M,N,P,Q$ locate on a circle - named $(I)$ 2. $MN\cap PQ$ at $J$. Prove that $IJ$ passed through a fixed line when $P,Q$ changed

2005 Georgia Team Selection Test, 11

Tags: geometry , rhombus , geometry proposed

On the sides $ AB, BC, CD$ and $ DA$ of the rhombus $ ABCD$, respectively, are chosen points $ E, F, G$ and $ H$ so, that $ EF$ and $ GH$ touch the incircle of the rhombus. Prove that the lines $ EH$ and $ FG$ are parallel.

2014 Dutch BxMO/EGMO TST, 3

Tags: geometry proposed , geometry

In triangle $ABC$, $I$ is the centre of the incircle. There is a circle tangent to $AI$ at $I$ which passes through $B$. This circle intersects $AB$ once more in $P$ and intersects $BC$ once more in $Q$. The line $QI$ intersects $AC$ in $R$. Prove that $|AR|\cdot |BQ|=|P I|^2$

2002 China Team Selection Test, 1

Tags: ratio , geometry , angle bisector , geometry proposed

Let $E$ and $F$ be the intersections of opposite sides of a convex quadrilateral $ABCD$. The two diagonals meet at $P$. Let $O$ be the foot of the perpendicular from $P$ to $EF$. Show that $\angle BOC=\angle AOD$.

2001 Flanders Math Olympiad, 2

Tags: geometry , algebra , system of equations , geometry proposed

Consider a triangle and 2 lines that each go through a corner and intersects the opposing segment, such that the areas are as on the attachment. Find the "?"

2008 Sharygin Geometry Olympiad, 24

Tags: inequalities , geometry , 3D geometry , tetrahedron , geometry proposed , geometry unsolved

(I.Bogdanov, 11) Let $ h$ be the least altitude of a tetrahedron, and $ d$ the least distance between its opposite edges. For what values of $ t$ the inequality $ d>th$ is possible?

2007 Moldova Team Selection Test, 1

Tags: conics , parabola , symmetry , ellipse , hyperbola , geometry proposed , geometry

Show that the plane cannot be represented as the union of the inner regions of a finite number of parabolas.

2002 Tournament Of Towns, 4

Tags: geometry proposed , geometry

Quadrilateral $ABCD$ is circumscribed about a circle $\Gamma$ and $K,L,M,N$ are points of tangency of sides $AB,BC,CD,DA$ with $\Gamma$ respectively. Let $S\equiv KM\cap LN$. If quadrilateral $SKBL$ is cyclic then show that $SNDM$ is also cyclic.

2010 Sharygin Geometry Olympiad, 23

Tags: geometry proposed , geometry

A cyclic hexagon $ABCDEF$ is such that $AB \cdot CF= 2BC \cdot FA, CD \cdot EB = 2 DE \cdot BC$ and $EF \cdot AD = 2FA \cdot DE.$ Prove that the lines $AD, BE$ and $CF$ are concurrent.

2010 ISI B.Stat Entrance Exam, 7

Tags: geometry , geometry proposed

Consider a rectangular sheet of paper $ABCD$ such that the lengths of $AB$ and $AD$ are respectively $7$ and $3$ centimetres. Suppose that $B'$ and $D'$ are two points on $AB$ and $AD$ respectively such that if the paper is folded along $B'D'$ then $A$ falls on $A'$ on the side $DC$. Determine the maximum possible area of the triangle $AB'D'$.

2011 CentroAmerican, 2

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

In a scalene triangle $ABC$, $D$ is the foot of the altitude through $A$, $E$ is the intersection of $AC$ with the bisector of $\angle ABC$ and $F$ is a point on $AB$. Let $O$ the circumcenter of $ABC$ and $X=AD\cap BE$, $Y=BE\cap CF$, $Z=CF \cap AD$. If $XYZ$ is an equilateral triangle, prove that one of the triangles $OXY$, $OYZ$, $OZX$ must be equilateral.

1991 Balkan MO, 1

Tags: geometry , geometric transformation , reflection , parallelogram , trapezoid , geometry proposed

Let $ABC$ be an acute triangle inscribed in a circle centered at $O$. Let $M$ be a point on the small arc $AB$ of the triangle's circumcircle. The perpendicular dropped from $M$ on the ray $OA$ intersects the sides $AB$ and $AC$ at the points $K$ and $L$, respectively. Similarly, the perpendicular dropped from $M$ on the ray $OB$ intersects the sides $AB$ and $BC$ at $N$ and $P$, respectively. Assume that $KL=MN$. Find the size of the angle $\angle{MLP}$ in terms of the angles of the triangle $ABC$.

1979 IMO Longlists, 22

Tags: geometry , perpendicular bisector , geometry proposed

Consider two quadrilaterals $ABCD$ and $A'B'C'D'$ in an affine Euclidian plane such that $AB = A'B', BC = B'C', CD = C'D'$, and $DA = D'A'$. Prove that the following two statements are true: [b](a)[/b] If the diagonals $BD$ and $AC$ are mutually perpendicular, then the diagonals $B'D'$ and $A'C'$ are also mutually perpendicular. [b](b)[/b] If the perpendicular bisector of $BD$ intersects $AC$ at $M$, and that of $B'D'$ intersects $A'C'$ at $M'$, then $\frac{\overline{MA}}{\overline{MC}}=\frac{\overline{M'A'}}{\overline{M'C'}}$ (if $MC = 0$ then $M'C' = 0$).

2006 Moldova National Olympiad, 11.3

Tags: geometry , 3D geometry , pyramid , ratio , geometry proposed

Let $ABCDE$ be a right quadrangular pyramid with vertex $E$ and height $EO$. Point $S$ divides this height in the ratio $ES: SO=m$. In which ratio does the plane $(ABC)$ divide the lateral area of the pyramid.

2011 ELMO Shortlist, 1

Tags: geometry , analytic geometry , geometry proposed

Let $ABCD$ be a convex quadrilateral. Let $E,F,G,H$ be points on segments $AB$, $BC$, $CD$, $DA$, respectively, and let $P$ be the intersection of $EG$ and $FH$. Given that quadrilaterals $HAEP$, $EBFP$, $FCGP$, $GDHP$ all have inscribed circles, prove that $ABCD$ also has an inscribed circle. [i]Evan O'Dorney.[/i]

1995 Irish Math Olympiad, 3

Tags: inequalities , geometry proposed , geometry

Points $ A,X,D$ lie on a line in this order, point $ B$ is on the plane such that $ \angle ABX>120^{\circ}$, and point $ C$ is on the segment $ BX$. Prove the inequality: $ 2AD \ge \sqrt{3} (AB\plus{}BC\plus{}CD)$.

2009 ITAMO, 2

Tags: geometry , geometric transformation , rotation , geometry proposed

$ABCD$ is a square with centre $O$. Two congruent isosceles triangle $BCJ$ and $CDK$ with base $BC$ and $CD$ respectively are constructed outside the square. let $M$ be the midpoint of $CJ$. Show that $OM$ and $BK$ are perpendicular to each other.

Indonesia MO Shortlist - geometry, g1.1

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

Given triangle $ ABC$. Points $ D,E,F$ outside triangle $ ABC$ are chosen such that triangles $ ABD$, $ BCE$, and $ CAF$ are equilateral triangles. Prove that cicumcircles of these three triangles are concurrent.

2024 Francophone Mathematical Olympiad, 3

Tags: geometry , geometry proposed

Let $ABC$ be an acute triangle with $AB<AC$ and let $O$ be its circumcenter. Let $D$ be a point on the segment $AC$ such that $AB=AD$. Let $E$ be the intersection of the line $AB$ with the perpendicular line to $AO$ through $D$. Let $F$ be the intersection of the perpendicular line to $OC$ through $C$ with the line parallel to $AC$ and passing through $E$. Finally, let the lines $CE$ and $DF$ intersect in $G$. Show that $AG$ and $BF$ are parallel.