Olympiad problems

2001 All-Russian Olympiad, 4

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

A sphere with center on the plane of the face $ABC$ of a tetrahedron $SABC$ passes through $A$, $B$ and $C$, and meets the edges $SA$, $SB$, $SC$ again at $A_1$, $B_1$, $C_1$, respectively. The planes through $A_1$, $B_1$, $C_1$ tangent to the sphere meet at $O$. Prove that $O$ is the circumcenter of the tetrahedron $SA_1B_1C_1$.

1984 IMO Longlists, 39

Tags: geometry , geometry unsolved

Let $ABC$ be an isosceles triangle, $AB = AC, \angle A = 20^{\circ}$. Let $D$ be a point on $AB$, and $E$ a point on $AC$ such that $\angle ACD = 20^{\circ}$ and $\angle ABE = 30^{\circ}$. What is the measure of the angle $\angle CDE$?

2007 Estonia National Olympiad, 2

Tags: geometry unsolved , geometry

Two medians drawn from vertices A and B of triangle ABC are perpendicular. Prove that side AB is the shortest side of ABC.

2015 Peru IMO TST, 8

Tags: geometry , geometry unsolved , Inversion , circles , tangent

Let $I$ be the incenter of the $ABC$ triangle. The circumference that passes through $I$ and has center in $A$ intersects the circumscribed circumference of the $ABC$ triangle at points $M$ and $N$. Prove that the line $MN$ is tangent to the inscribed circle of the $ABC$ triangle.

2011 Estonia Team Selection Test, 1

Tags: geometric transformation , geometry unsolved , geometry

Two circles lie completely outside each other.Let $A$ be the point of intersection of internal common tangents of the circles and let $K$ be the projection of this point onto one of their external common tangents.The tangents,different from the common tangent,to the circles through point $K$ meet the circles at $M_1$ and $M_2$.Prove that the line $AK$ bisects angle $M_1 KM_2$.

1998 All-Russian Olympiad, 7

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

A tetrahedron $ABCD$ has all edges of length less than $100$, and contains two nonintersecting spheres of diameter $1$. Prove that it contains a sphere of diameter $1.01$.

JBMO Geometry Collection, 2011

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

Let $ABCD$ be a convex quadrilateral and points $E$ and $F$ on sides $AB,CD$ such that \[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\] If $S$ is the area of $AEFD$ show that ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$

2013 Indonesia MO, 2

Tags: geometry , circumcircle , geometry unsolved

Let $ABC$ be an acute triangle and $\omega$ be its circumcircle. The bisector of $\angle BAC$ intersects $\omega$ at [another point] $M$. Let $P$ be a point on $AM$ and inside $\triangle ABC$. Lines passing $P$ that are parallel to $AB$ and $AC$ intersects $BC$ on $E, F$ respectively. Lines $ME, MF$ intersects $\omega$ at points $K, L$ respectively. Prove that $AM, BL, CK$ are concurrent.

2013 Baltic Way, 11

Tags: geometry , circumcircle , trigonometry , geometry unsolved

In an acute triangle $ABC$ with $AC > AB$, let $D$ be the projection of $A$ on $BC$, and let $E$ and $F$ be the projections of $D$ on $AB$ and $AC$, respectively. Let $G$ be the intersection point of the lines $AD$ and $EF$. Let $H$ be the second intersection point of the line $AD$ and the circumcircle of triangle $ABC$. Prove that \[AG \cdot AH=AD^2\]

1978 IMO Longlists, 21

Tags: geometry , analytic geometry , geometry unsolved

A circle touches the sides $AB,BC, CD,DA$ of a square at points $K,L,M,N$ respectively, and $BU, KV$ are parallel lines such that $U$ is on $DM$ and $V$ on $DN$. Prove that $UV$ touches the circle.

2006 Tuymaada Olympiad, 3

Tags: geometry , geometric transformation , homothety , power of a point , radical axis , geometry unsolved

A line $d$ is given in the plane. Let $B\in d$ and $A$ another point, not on $d$, and such that $AB$ is not perpendicular on $d$. Let $\omega$ be a variable circle touching $d$ at $B$ and letting $A$ outside, and $X$ and $Y$ the points on $\omega$ such that $AX$ and $AY$ are tangent to the circle. Prove that the line $XY$ passes through a fixed point. [i]Proposed by F. Bakharev [/i]

2014 Germany Team Selection Test, 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 \[ AN \cdot NC = CD \cdot BN. \]

2007 Vietnam National Olympiad, 3

Tags: analytic geometry , geometry , conics , hyperbola , geometry unsolved

Let B,C be fixed points and A be roving point. Let H, G be orthecentre and centroid of triagle ABC. Known midpoint of HG lies on BC, find locus of A

1999 Hong kong National Olympiad, 2

Tags: geometry , incenter , trigonometry , geometry unsolved

Let $I$ be the incentre and $O$ the circumcentre of a non-equilateral triangle $ABC$. Prove that $\angle AIO \le 90^{\circ}$ if and only if $2BC\le AB+AC$.

2007 Finnish National High School Mathematics Competition, 3

Tags: combinatorial geometry , geometry unsolved , geometry

There are five points in the plane, no three of which are collinear. Show that some four of these points are the vertices of a convex quadrilateral.

2010 Indonesia TST, 3

Tags: geometry unsolved , geometry

Given a non-isosceles triangle $ABC$ with incircle $k$ with center $S$. $k$ touches the side $BC,CA,AB$ at $P,Q,R$ respectively. The line $QR$ and line $BC$ intersect at $M$. A circle which passes through $B$ and $C$ touches $k$ at $N$. The circumcircle of triangle $MNP$ intersects $AP$ at $L$. Prove that $S,L,M$ are collinear.

2011 Kyrgyzstan National Olympiad, 5

Tags: geometry unsolved , geometry

Points $M$ and $N$ are chosen on sides $AB$ and $BC$,respectively, in a triangle $ABC$, such that point $O$ is interserction of lines $CM$ and $AN$. Given that $AM+AN=CM+CN$. Prove that $AO+AB=CO+CB$.

1993 Turkey MO (2nd round), 5

Tags: geometry unsolved , geometry

Prove that we can draw a line (by a ruler and a compass) from a vertice of a convex quadrilateral such that, the line divides the quadrilateral to two equal areas.

2001 Cono Sur Olympiad, 1

Tags: geometry , analytic geometry , geometry unsolved

A polygon of area $S$ is contained inside a square of side length $a$. Show that there are two points of the polygon that are a distance of at least $S/a$ apart.

2013 European Mathematical Cup, 4

Tags: geometry , circumcircle , geometry unsolved

Given a triangle $ABC$ let $D$, $E$, $F$ be orthogonal projections from $A$, $B$, $C$ to the opposite sides respectively. Let $X$, $Y$, $Z$ denote midpoints of $AD$, $BE$, $CF$ respectively. Prove that perpendiculars from $D$ to $YZ$, from $E$ to $XZ$ and from $F$ to $XY$ are concurrent.

2009 Mexico National Olympiad, 1

Tags: geometry unsolved , geometry

In $\triangle ABC$, let $D$ be the foot of the altitude from $A$ to $BC$. A circle centered at $D$ with radius $AD$ intersects lines $AB$ and $AC$ at $P$ and $Q$, respectively. Show that $\triangle AQP\sim\triangle ABC$.

2012 Sharygin Geometry Olympiad, 21

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

Two perpendicular lines pass through the orthocenter of an acute-angled triangle. The sidelines of the triangle cut on each of these lines two segments: one lying inside the triangle and another one lying outside it. Prove that the product of two internal segments is equal to the product of two external segments. [i]Nikolai Beluhov and Emil Kolev[/i]

2011 Poland - Second Round, 1

Tags: geometry , geometric transformation , rotation , geometry unsolved

Points $A,B,C,D,E,F$ lie in that order on semicircle centered at $O$, we assume that $AD=BE=CF$. $G$ is a common point of $BE$ and $AD$, $H$ is a common point of $BE$ and $CD$. Prove that: \[\angle AOC=2\angle GOH.\]

2008 Romanian Master of Mathematics, 1

Tags: conics , geometry unsolved , geometry

Let $ ABC$ be an equilateral triangle and $ P$ in its interior. The distances from $ P$ to the triangle's sides are denoted by $ a^2, b^2,c^2$ respectively, where $ a,b,c>0$. Find the locus of the points $ P$ for which $ a,b,c$ can be the sides of a non-degenerate triangle.

2002 Austrian-Polish Competition, 3

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

Let $ABCD$ be a tetrahedron and let $S$ be its center of gravity. A line through $S$ intersects the surface of $ABCD$ in the points $K$ and $L$. Prove that \[\frac{1}{3}\leq \frac{KS}{LS}\leq 3\]