Olympiad problems

2011 Sharygin Geometry Olympiad, 1

Does a convex heptagon exist which can be divided into 2011 equal triangles?

2004 Poland - Second Round, 2

Tags: geometry unsolved , geometry

In convex hexagon $ ABCDEF$ all sides have equal length and $ \angle A\plus{}\angle C\plus{}\angle E\equal{}\angle B\plus{}\angle D\plus{}\angle F$. Prove that the diagonals $ AD,BE,CF$ are concurrent.

2003 Tournament Of Towns, 6

Tags: geometry unsolved , geometry

An ant crawls on the outer surface of the box in a shape of rectangular parallelepiped. From ant’s point of view, the distance between two points on a surface is defined by the length of the shortest path ant need to crawl to reach one point from the other. Is it true that if ant is at vertex then from ant’s point of view the opposite vertex be the most distant point on the surface?

Estonia Open Senior - geometry, 2007.2.5

Tags: geometry , circumcircle , inradius , trigonometry , geometry unsolved

Consider triangles whose each side length squared is a rational number. Is it true that (a) the square of the circumradius of every such triangle is rational; (b) the square of the inradius of every such triangle is rational?

1999 China Team Selection Test, 1

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

A circle is tangential to sides $AB$ and $AD$ of convex quadrilateral $ABCD$ at $G$ and $H$ respectively, and cuts diagonal $AC$ at $E$ and $F$. What are the necessary and sufficient conditions such that there exists another circle which passes through $E$ and $F$, and is tangential to $DA$ and $DC$ extended?

2004 Tournament Of Towns, 4

Tags: geometry unsolved , geometry

We have a circle and a line which does not intersect the circle. Using only compass and straightedge, construct a square whose two adjacent vertices are on the circle, and two other vertices are on the given line (it is known that such a square exists).

2012 Poland - Second Round, 2

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

Prove that for tetrahedron $ABCD$; vertex $D$, center of insphere and centroid of $ABCD$ are collinear iff areas of triangles $ABD,BCD,CAD$ are equal.

2012 Indonesia TST, 3

Tags: quadratics , geometry , rectangle , vector , geometry unsolved

The [i]cross[/i] of a convex $n$-gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a $3 \times 4$ rectangle is $\sqrt{ \dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \dfrac{5}{3} \sqrt{6}$. Suppose $S$ is a dodecagon ($12$-gon) inscribed in a unit circle. Find the greatest possible cross of $S$.

2008 Sharygin Geometry Olympiad, 4

Tags: geometry , circumcircle , angle bisector , geometry unsolved

(F.Nilov, A.Zaslavsky) Let $ CC_0$ be a median of triangle $ ABC$; the perpendicular bisectors to $ AC$ and $ BC$ intersect $ CC_0$ in points $ A_c$, $ B_c$; $ C_1$ is the common point of $ AA_c$ and $ BB_c$. Points $ A_1$, $ B_1$ are defined similarly. Prove that circle $ A_1B_1C_1$ passes through the circumcenter of triangle $ ABC$.

2018 IFYM, Sozopol, 5

Tags: geometry , geometry unsolved

On the extension of the heights $AH_1$ and $BH_2$ of an acute $\triangle ABC$, after points $H_1$ and $H_2$, are chosen points $M$ and $N$ in such way that $\angle MCB = \angle NCA = 30^\circ$. We denote with $C_1$ the intersection point of the lines $MB$ and $NA$. Analogously we define $A_1$ and $B_1$. Prove that the straight lines $AA_1$, $BB_1$, and $CC_1$ intersect in one point.

1991 Kurschak Competition, 2

Tags: geometry unsolved , geometry

A convex polyhedron has two triangle and three quadrilateral faces. Connect every vertex of one of the triangle faces with the intersection point of the diagonals in the quadrilateral face opposite to it. Show that the resulting three lines are concurrent.

2006 China Northern MO, 1

Tags: projective geometry , geometry unsolved , geometry

$AB$ is the diameter of circle $O$, $CD$ is a non-diameter chord that is perpendicular to $AB$. Let $E$ be the midpoint of $OC$, connect $AE$ and extend it to meet the circle at point $P$. Let $DP$ and $BC$ meet at $F$. Prove that $F$ is the midpoint of $BC$.

1978 IMO Longlists, 49

Tags: geometry unsolved , geometry

Let $A,B,C,D$ be four arbitrary distinct points in space. $(a)$ Prove that using the segments $AB +CD, AC +BD$ and $AD +BC$, it is always possible to construct a triangle $T$ that is non-degenerate and has no obtuse angle. $(b)$ What should these four points satisfy in order for the triangle $T$ to be right-angled?

2016 Germany Team Selection Test, 1

Tags: Concyclic , geometry unsolved , geometry , circle , circles

The two circles $\Gamma_1$ and $\Gamma_2$ with the midpoints $O_1$ resp. $O_2$ intersect in the two distinct points $A$ and $B$. A line through $A$ meets $\Gamma_1$ in $C \neq A$ and $\Gamma_2$ in $D \neq A$. The lines $CO_1$ and $DO_2$ intersect in $X$. Prove that the four points $O_1,O_2,B$ and $X$ are concyclic.

2014 Argentina Cono Sur TST, 3

Tags: geometry , geometry unsolved

All diagonals of a convex pentagon are drawn, dividing it in one smaller pentagon and $10$ triangles. Find the maximum number of triangles with the same area that may exist in the division.

2010 Sharygin Geometry Olympiad, 24

Tags: geometry unsolved , geometry

Let us have a line $\ell$ in the space and a point $A$ not lying on $\ell.$ For an arbitrary line $\ell'$ passing through $A$, $XY$ ($Y$ is on $\ell'$) is a common perpendicular to the lines $\ell$ and $\ell'.$ Find the locus of points $Y.$

1999 China Team Selection Test, 1

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

A circle is tangential to sides $AB$ and $AD$ of convex quadrilateral $ABCD$ at $G$ and $H$ respectively, and cuts diagonal $AC$ at $E$ and $F$. What are the necessary and sufficient conditions such that there exists another circle which passes through $E$ and $F$, and is tangential to $DA$ and $DC$ extended?

2005 Junior Balkan Team Selection Tests - Romania, 9

Tags: inequalities , geometry unsolved , geometry

Let $ABC$ be a triangle with $BC>CA>AB$ and let $G$ be the centroid of the triangle. Prove that \[ \angle GCA+\angle GBC<\angle BAC<\angle GAC+\angle GBA . \] [i]Dinu Serbanescu[/i]

2004 Federal Competition For Advanced Students, P2, 6

Tags: geometry , geometry unsolved

Over the sides of an equilateral triangle with area $ 1$ are triangles with the opposite angle $ 60^{\circ}$ to each side drawn outside of the triangle. The new corners are $ P$, $ Q$ and $ R$. (and the new triangles $ APB$, $ BQC$ and $ ARC$) 1)What is the highest possible area of the triangle $ PQR$? 2)What is the highest possible area of the triangle whose vertexes are the midpoints of the inscribed circles of the triangles $ APB$, $ BQC$ and $ ARC$?

1989 IMO Longlists, 65

Tags: geometry unsolved , geometry

Let $ ABCD$ be a quadrilateral inscribed in a circle of radius $ AB$ such that $ BC \equal{} a, CD \equal{} b,$ $ DA \equal{} \frac{3 \sqrt{3} \minus{} 1}{2} \cdot a$ For each point $ M$ on the semicircle with radius $ AB$ not containing $ C$ and $ D,$ denote by $ h_1, h_2, h_3$ the distances from $ M$ to the straight lines (sides) $ BC, CD,$ and $ DA.$ Find the maximum of $ h_1 \plus{} h_2 \plus{} h_3.$

2010 China Team Selection Test, 1

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

Let $\triangle ABC$ be an acute triangle with $AB>AC$, let $I$ be the center of the incircle. Let $M,N$ be the midpoint of $AC$ and $AB$ respectively. $D,E$ are on $AC$ and $AB$ respectively such that $BD\parallel IM$ and $CE\parallel IN$. A line through $I$ parallel to $DE$ intersects $BC$ in $P$. Let $Q$ be the projection of $P$ on line $AI$. Prove that $Q$ is on the circumcircle of $\triangle ABC$.

2009 Tuymaada Olympiad, 2

Tags: geometry , trapezoid , geometric transformation , homothety , angle bisector , geometry unsolved

$ M$ is the midpoint of base $ BC$ in a trapezoid $ ABCD$. A point $ P$ is chosen on the base $ AD$. The line $ PM$ meets the line $ CD$ at a point $ Q$ such that $ C$ lies between $ Q$ and $ D$. The perpendicular to the bases drawn through $ P$ meets the line $ BQ$ at $ K$. Prove that $ \angle QBC \equal{} \angle KDA$. [i]Proposed by S. Berlov[/i]

1995 Kurschak Competition, 3

Tags: geometry unsolved , geometry

Points $A$, $B$, $C$, $D$ are such that no three of them are collinear. Let $E=AB\cap CD$ and $F=BC\cap DA$. Let $k_1$, $k_2$ and $k_3$ denote the circles with diameter $\overline{AC}$, $\overline{BD}$ and $\overline{EF}$, respectively. Prove that either $k_1,k_2,k_3$ pass through one point, or no two of them intersect.

2012 Sharygin Geometry Olympiad, 4

Tags: analytic geometry , geometry , parallelogram , ratio , perpendicular bisector , congruent triangles , geometry unsolved

Given triangle $ABC$. Point $M$ is the midpoint of side $BC$, and point $P$ is the projection of $B$ to the perpendicular bisector of segment $AC$. Line $PM$ meets $AB$ in point $Q$. Prove that triangle $QPB$ is isosceles.

1980 Canada National Olympiad, 5

Tags: geometry , perimeter , geometry unsolved

A parallelepiped has the property that all cross sections, which are parallel to any fixed face $F$, have the same perimeter as $F$. Determine whether or not any other polyhedron has this property. Typesetter's Note: I believe that proof of existence or non-existence suffices.