Olympiad problems

2006 MOP Homework, 4

Let $ABCD$ be a tetrahedron and let $H_{a},H_{b},H_{c},H_{d}$ be the orthocenters of triangles $BCD,CDA,DAB,ABC$, respectively. Prove that lines $AH_{a},BH_{b},CH_{c}, DH_{d}$ are concurrent if and only if $AB^2 + CD^2 = AC^2 + BD^2 = AD^2 + BC^2$

2003 Tournament Of Towns, 1

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

A triangular pyramid $ABCD$ is given. Prove that $\frac Rr > \frac ah$, where $R$ is the radius of the circumscribed sphere, $r$ is the radius of the inscribed sphere, $a$ is the length of the longest edge, $h$ is the length of the shortest altitude (from a vertex to the opposite face).

2008 Mexico National Olympiad, 3

Tags: geometry , circumcircle , power of a point , radical axis , geometry unsolved

The internal angle bisectors of $A$, $B$, and $C$ in $\triangle ABC$ concur at $I$ and intersect the circumcircle of $\triangle ABC$ at $L$, $M$, and $N$, respectively. The circle with diameter $IL$ intersects $BC$ at $D$ and $E$; the circle with diameter $IM$ intersects $CA$ at $F$ and $G$; the circle with diameter $IN$ intersects $AB$ at $H$ and $J$. Show that $D$, $E$, $F$, $G$, $H$, and $J$ are concyclic.

2010 Contests, 1

Tags: geometry unsolved , geometry

Let $A$ and $B$ be two fixed points of a given circle and $XY$ a diameter of this circle. Find the locus of the intersection points of lines $AX$ and $BY$ . ($BY$ is not a diameter of the circle). Albanian National Mathematical Olympiad 2010---12 GRADE Question 1.

1979 IMO Longlists, 48

Tags: geometry unsolved , geometry

In the plane a circle $C$ of unit radius is given. For any line $l$, a number $s(l)$ is defined in the following way: If $l$ and $C$ intersect in two points, $s(l)$ is their distance; otherwise, $s(l) = 0$. Let $P$ be a point at distance $r$ from the center of $C$. One defines $M(r)$ to be the maximum value of the sum $s(m) + s(n)$, where $m$ and $n$ are variable mutually orthogonal lines through $P$. Determine the values of $r$ for which $M(r) > 2$.

1993 All-Russian Olympiad, 2

Tags: geometry , parallelogram , geometry unsolved

A convex quadrilateral intersects a circle at points $A_1,A_2,B_1,B_2,C_1,C_2,D_1,$ and $D_2$. (Note that for some letter $N$, points $N_1$ and $N_2$ are on one side of the quadrilateral. Also, the points lie in that specific order on the circle.) Prove that if $A_1B_2=B_1C_2=C_1D_2= D_1A_2$, then quadrilateral formed by these four segments is cyclic.

2006 Korea - Final Round, 2

Tags: geometry unsolved , geometry

In a convex hexagon $ABCDEF$ triangles $ABC , CDE , EFA$ are similar. Find conditions on these triangles under which triangle $ACE$ is equilateral if and only if so is $BDF.$

2014 Postal Coaching, 2

Tags: geometry unsolved , geometry

Suppose $ABCD$ is a convex quadrilateral.Points $P,Q,R$ and $S$ are four points on the line segments $AB,BC,CD$ and $DA$ respectively.The line segments $PR$ and $QS$ meet at $T$.Suppose that each of the quadrilaterals $APTS,BQTP,CRTQ$ and $DSTR$ have an incircle.Prove that the quadrilateral $ABCD$ also has an incircle.

1980 IMO, 21

Tags: geometry unsolved , geometry

Let $ABCDEFGH$ be the rectangular parallelepiped where $ABCD$ and $EFGH$ are squares and the edges $AE,BF,CG,DH$ are all perpendicular to the squares. Prove that if the $12$ edges of the parallelepiped have integer lengths, the internal diagonal $AG$ and the face diagonal $AF$ cannot both have integer length.

2011 Postal Coaching, 3

Tags: geometry , circumcircle , conics , geometry unsolved

Construct a triangle, by straight edge and compass, if the three points where the extensions of the medians intersect the circumcircle of the triangle are given.

2013 Benelux, 3

Tags: geometry , circumcircle , ratio , power of a point , geometry unsolved

Let $\triangle ABC$ be a triangle with circumcircle $\Gamma$, and let $I$ be the center of the incircle of $\triangle ABC$. The lines $AI$, $BI$ and $CI$ intersect $\Gamma$ in $D \ne A$, $E \ne B$ and $F \ne C$. The tangent lines to $\Gamma$ in $F$, $D$ and $E$ intersect the lines $AI$, $BI$ and $CI$ in $R$, $S$ and $T$, respectively. Prove that \[\vert AR\vert \cdot \vert BS\vert \cdot \vert CT\vert = \vert ID\vert \cdot \vert IE\vert \cdot \vert IF\vert.\]

2003 China Girls Math Olympiad, 1

Tags: inequalities , Cauchy Inequality , geometry unsolved , geometry

Let $ ABC$ be a triangle. Points $ D$ and $ E$ are on sides $ AB$ and $ AC,$ respectively, and point $ F$ is on line segment $ DE.$ Let $ \frac {AD}{AB} \equal{} x,$ $ \frac {AE}{AC} \equal{} y,$ $ \frac {DF}{DE} \equal{} z.$ Prove that (1) $ S_{\triangle BDF} \equal{} (1 \minus{} x)y S_{\triangle ABC}$ and $ S_{\triangle CEF} \equal{} x(1 \minus{} y) (1 \minus{} z)S_{\triangle ABC};$ (2) $ \sqrt [3]{S_{\triangle BDF}} \plus{} \sqrt [3]{S_{\triangle CEF}} \leq \sqrt [3]{S_{\triangle ABC}}.$

1999 Tuymaada Olympiad, 1

Tags: geometry , geometric transformation , reflection , trigonometry , geometry unsolved

In the triangle $ABC$ we have $\angle ABC=100^\circ$, $\angle ACB=65^\circ$, $M\in AB$, $N\in AC$, and $\angle MCB=55^\circ$, $\angle NBC=80^\circ$. Find $\angle NMC$. [i]St.Petersburg folklore[/i]

1992 IMTS, 1

Tags: geometry , geometry unsolved

Nine lines, parallel to the base of a triangle, divide the other sides into 10 equal segments and the area into 10 distinct parts. Find the area of the original triangle, if the area of the largest of these parts is 76.

2006 India IMO Training Camp, 3

Tags: geometry unsolved , geometry

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 $ABC$.

2007 Croatia Team Selection Test, 3

Tags: search , geometry unsolved , geometry

Let $ABC$ be a triangle such that $|AC|>|AB|$. Let $X$ be on line $AB$ (closer to $A$) such that $|BX|=|AC|$ and let $Y$ be on the segment $AC$ such that $|CY|=|AB|$. Intersection of lines $XY$ and bisector of $BC$ is point $P$. Prove that $\angle BPC+\angle BAC = 180^\circ$.

2002 Iran Team Selection Test, 3

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

A "[i]2-line[/i]" is the area between two parallel lines. Length of "2-line" is distance of two parallel lines. We have covered unit circle with some "2-lines". Prove sum of lengths of "2-lines" is at least 2.

1972 IMO Longlists, 14

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

$(a)$ A plane $\pi$ passes through the vertex $O$ of the regular tetrahedron $OPQR$. We define $p, q, r$ to be the signed distances of $P,Q,R$ from $\pi$ measured along a directed normal to $\pi$. Prove that \[p^2 + q^2 + r^2 + (q - r)^2 + (r - p)^2 + (p - q)^2 = 2a^2,\] where $a$ is the length of an edge of a tetrahedron. $(b)$ Given four parallel planes not all of which are coincident, show that a regular tetrahedron exists with a vertex on each plane. [u]Note:[/u] Part $(b)$ is [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=49&t=60825&start=0]IMO 1972 Problem 6[/url]

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.

2011 Croatia Team Selection Test, 3

Tags: geometry , circumcircle , perpendicular bisector , projective geometry , power of a point , radical axis , geometry unsolved

Let $K$ and $L$ be the points on the semicircle with diameter $AB$. Denote intersection of $AK$ and $AL$ as $T$ and let $N$ be the point such that $N$ is on segment $AB$ and line $TN$ is perpendicular to $AB$. If $U$ is the intersection of perpendicular bisector of $AB$ an $KL$ and $V$ is a point on $KL$ such that angles $UAV$ and $UBV$ are equal. Prove that $NV$ is perpendicular to $KL$.

1989 IMO Longlists, 6

Tags: function , geometry , analytic geometry , geometry unsolved

Let $ E$ be the set of all triangles whose only points with integer coordinates (in the Cartesian coordinate system in space), in its interior or on its sides, are its three vertices, and let $ f$ be the function of area of a triangle. Determine the set of values $ f(E)$ of $ f.$

Estonia Open Senior - geometry, 2006.1.3

Tags: geometry unsolved , geometry

Let $ ABC$ be an acute triangle and choose points $ A_1, B_1$ and $ C_1$ on sides $ BC, CA$ and $ AB$, respectively. Prove that if the quadrilaterals $ ABA_1B_1, BCB_1C_1$ and $ CAC_1A_1$ are cyclic then their circumcentres lie on the sides of $ ABC$.

2004 China Girls Math Olympiad, 6

Tags: trigonometry , geometry unsolved , geometry

Given an acute triangle $ABC$ with $O$ as its circumcenter. Line $AO$ intersects $BC$ at $D$. Points $E$, $F$ are on $AB$, $AC$ respectively such that $A$, $E$, $D$, $F$ are concyclic. Prove that the length of the projection of line segment $EF$ on side $BC$ does not depend on the positions of $E$ and $F$.

2009 Germany Team Selection Test, 1

Tags: geometry , geometry unsolved

Let $ I$ be the incircle centre of triangle $ ABC$ and $ \omega$ be a circle within the same triangle with centre $ I.$ The perpendicular rays from $ I$ on the sides $ \overline{BC}, \overline{CA}$ and $ \overline{AB}$ meets $ \omega$ in $ A', B'$ and $ C'.$ Show that the three lines $ AA', BB'$ and $ CC'$ have a common point.

1973 IMO Longlists, 5

Tags: geometry , parallelogram , geometry unsolved

Given a ball $K$. Find the locus of the vertices $A$ of all parallelograms $ABCD$ such that $ AC \leq BD$, and the diagonal $BD$ lies completely inside the ball $K$.