Olympiad problems

2002 Tournament Of Towns, 2

Can any triangle be cut into four convex figures: a triangle, a quadrilateral, a pentagon, a hexagon?

1990 Hungary-Israel Binational, 2

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

Let $ ABC$ be a triangle where $ \angle ACB\equal{}90^{\circ}$. Let $ D$ be the midpoint of $ BC$ and let $ E$, and $ F$ be points on $ AC$ such that $ CF\equal{}FE\equal{}EA$. The altitude from $ C$ to the hypotenuse $ AB$ is $ CG$, and the circumcentre of triangle $ AEG$ is $ H$. Prove that the triangles $ ABC$ and $ HDF$ are similar.

2010 Sharygin Geometry Olympiad, 21

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

A given convex quadrilateral $ABCD$ is such that $\angle ABD + \angle ACD > \angle BAC + \angle BDC.$ Prove that \[S_{ABD}+S_{ACD} > S_{BAC}+S_{BDC}.\]

2005 Germany Team Selection Test, 3

Tags: geometry , inequalities , geometry proposed

Let $ABC$ be a triangle with area $S$, and let $P$ be a point in the plane. Prove that $AP+BP+CP\geq 2\sqrt[4]{3}\sqrt{S}$.

2009 Bosnia Herzegovina Team Selection Test, 1

Tags: ratio , exterior angle , geometry proposed , geometry

Denote by $M$ and $N$ feets of perpendiculars from $A$ to angle bisectors of exterior angles at $B$ and $C,$ in triangle $\triangle ABC.$ Prove that the length of segment $MN$ is equal to semiperimeter of triangle $\triangle ABC.$

2009 Canadian Mathematical Olympiad Qualification Repechage, 6

Tags: geometry proposed , geometry

Triangle $ABC$ is right-angled at $C$. $AQ$ is drawn parallel to $BC$ with $Q$ and $B$ on opposite sides of $AC$ so that when $BQ$ is drawn, intersecting $AC$ at $P$, we have $PQ = 2AB$. Prove that $\angle ABC = 3\angle PBC$.

2010 JBMO Shortlist, 4

Tags: geometry , circumcircle , trigonometry , perpendicular bisector , angle bisector , geometry proposed

Let $AL$ and $BK$ be angle bisectors in the non-isosceles triangle $ABC$ ($L$ lies on the side $BC$, $K$ lies on the side $AC$). The perpendicular bisector of $BK$ intersects the line $AL$ at point $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK$. Prove that $LN = NA$.

1995 All-Russian Olympiad, 4

Tags: geometry proposed , geometry

Prove that if all angles of a convex $n$-gon are equal, then there are at least two of its sides that are not longer than their adjacent sides. [i]A. Berzin’sh, O. Musin[/i]

2007 Romania Team Selection Test, 3

Tags: geometry , trigonometry , geometry proposed

Let $ABCDE$ be a convex pentagon, such that $AB=BC$, $CD=DE$, $\angle B+\angle D=180^{\circ}$, and it's area is $\sqrt2$. a) If $\angle B=135^{\circ}$, find the length of $[BD]$. b) Find the minimum of the length of $[BD]$.

2013 India National Olympiad, 5

Tags: geometry , rectangle , geometry proposed

In an acute triangle $ABC,$ let $O,G,H$ be its circumcentre, centroid and orthocenter. Let $D\in BC, E\in CA$ and $OD\perp BC, HE\perp CA.$ Let $F$ be the midpoint of $AB.$ If the triangles $ODC, HEA, GFB$ have the same area, find all the possible values of $\angle C.$

1982 Canada National Olympiad, 5

Tags: geometry , 3D geometry , tetrahedron , vector , geometry proposed

The altitudes of a tetrahedron $ABCD$ are extended externally to points $A'$, $B'$, $C'$, and $D'$, where $AA' = k/h_a$, $BB' = k/h_b$, $CC' = k/h_c$, and $DD' = k/h_d$. Here, $k$ is a constant and $h_a$ denotes the length of the altitude of $ABCD$ from vertex $A$, etc. Prove that the centroid of tetrahedron $A'B'C'D'$ coincides with the centroid of $ABCD$.

2010 Turkey Junior National Olympiad, 1

Tags: geometry , rectangle , ratio , circumcircle , power of a point , geometry proposed

A circle that passes through the vertex $A$ of a rectangle $ABCD$ intersects the side $AB$ at a second point $E$ different from $B.$ A line passing through $B$ is tangent to this circle at a point $T,$ and the circle with center $B$ and passing through $T$ intersects the side $BC$ at the point $F.$ Show that if $\angle CDF= \angle BFE,$ then $\angle EDF=\angle CDF.$

1998 Balkan MO, 3

Tags: geometry proposed , geometry

Let $\mathcal S$ denote the set of points inside or on the border of a triangle $ABC$, without a fixed point $T$ inside the triangle. Show that $\mathcal S$ can be partitioned into disjoint closed segemnts. [i]Yugoslavia[/i]

2010 Contests, 3

Tags: geometry , rectangle , geometry proposed

Two rectangles of unit area overlap to form a convex octagon. Show that the area of the octagon is at least $\dfrac {1} {2}$. [i]Kvant Magazine [/i]

1986 Iran MO (2nd round), 1

Tags: rotation , geometry , geometric transformation , reflection , symmetry , geometry proposed

$O$ is a point in the plane. Let $O'$ be an arbitrary point on the axis $Ox$ of the plane and let $M$ be an arbitrary point. Rotate $M$, $90^\circ$ clockwise around $O$ to get the point $M'$ and rotate $M$, $90^\circ$ anticlockwise around $O'$ to get the point $M''.$ Prove that the midpoint of the segment $MM''$ is a fixed point.

1995 Vietnam National Olympiad, 3

Tags: geometry proposed , geometry

Let a non-equilateral triangle $ ABC$ and $ AD,BE,CF$ are its altitudes. On the rays $ AD,BE,CF,$ respectively, let $ A',B',C'$ such that $ \frac {AA'}{AD} \equal{} \frac {BB'}{BE} \equal{} \frac {CC'}{CF} \equal{} k$. Find all values of $ k$ such that $ \triangle A'B'C'\sim\triangle ABC$ for any non-triangle $ ABC.$

1990 Turkey Team Selection Test, 1

Tags: geometry proposed , geometry

The circles $k_1, k_2, k_3$ with radii ($a>c>b$) $a,b,c$ are tangent to line $d$ at $A,B,C$, respectively. $k_1$ is tangent to $k_2$, and $k_2$ is tangent to $k_3$. The tangent line to $k_3$ at $E$ is parallel to $d$, and it meets $k_1$ at $D$. The line perpendicular to $d$ at $A$ meets line $EB$ at $F$. Prove that $AD=AF$.

2024 All-Russian Olympiad, 6

Tags: geometry , circumcircle , geometry proposed

The altitudes of an acute triangle $ABC$ with $AB<AC$ intersect at a point $H$, and $O$ is the center of the circumcircle $\Omega$. The segment $OH$ intersects the circumcircle of $BHC$ at a point $X$, different from $O$ and $H$. The circumcircle of $AOX$ intersects the smaller arc $AB$ of $\Omega$ at point $Y$. Prove that the line $XY$ bisects the segment $BC$. [i]Proposed by A. Tereshin[/i]

2002 Vietnam National Olympiad, 2

Tags: geometry , geometric transformation , reflection , geometry proposed

An isosceles triangle $ ABC$ with $ AB \equal{} AC$ is given on the plane. A variable circle $ (O)$ with center $ O$ on the line $ BC$ passes through $ A$ and does not touch either of the lines $ AB$ and $ AC$. Let $ M$ and $ N$ be the second points of intersection of $ (O)$ with lines $ AB$ and $ AC$, respectively. Find the locus of the orthocenter of triangle $ AMN$.

2000 239 Open Mathematical Olympiad, 7

Tags: geometry , incenter , circumcircle , Euler , angle bisector , perpendicular bisector , geometry proposed

The perpendicular bisectors of the sides AB and BC of a triangle ABC meet the lines BC and AB at the points X and Z, respectively. The angle bisectors of the angles XAC and ZCA intersect at a point B'. Similarly, define two points C' and A'. Prove that the points A', B', C' lie on one line through the incenter I of triangle ABC. [i]Extension:[/i] Prove that the points A', B', C' lie on the line OI, where O is the circumcenter and I is the incenter of triangle ABC. Darij

2003 Turkey Team Selection Test, 2

Tags: limit , inequalities , geometry proposed , geometry

Let $K$ be the intersection of the diagonals of a convex quadrilateral $ABCD$. Let $L\in [AD]$, $M \in [AC]$, $N \in [BC]$ such that $KL\parallel AB$, $LM\parallel DC$, $MN\parallel AB$. Show that \[\dfrac{Area(KLMN)}{Area(ABCD)} < \dfrac {8}{27}.\]

2011 Tokio University Entry Examination, 1

Tags: analytic geometry , geometry , trigonometry , geometry proposed

On the coordinate plane, let $C$ be a circle centered $P(0,\ 1)$ with radius 1. let $a$ be a real number $a$ satisfying $0<a<1$. Denote by $Q,\ R$ intersection points of the line $y=a(x+1) $ and $C$. (1) Find the area $S(a)$ of $\triangle{PQR}$. (2) When $a$ moves in the range of $0<a<1$, find the value of $a$ for which $S(a)$ is maximized. [i]2011 Tokyo University entrance exam/Science, Problem 1[/i]

2008 Sharygin Geometry Olympiad, 4

Tags: geometry , cyclic quadrilateral , geometry proposed

(D.Shnol, 8--9) The bisectors of two angles in a cyclic quadrilateral are parallel. Prove that the sum of squares of some two sides in the quadrilateral equals the sum of squares of two remaining sides.

2015 Brazil National Olympiad, 1

Tags: geometry proposed , geometry , circumcircle

Let $\triangle ABC$ be an acute-scalene triangle, and let $N$ be the center of the circle wich pass trough the feet of altitudes. Let $D$ be the intersection of tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$. Prove that $A$, $D$ and $N$ are collinear iff $\measuredangle BAC = 45º$.

2007 Tournament Of Towns, 2

Tags: geometry , cyclic quadrilateral , geometry proposed

Let $K, L, M$ and $N$ be the midpoints of the sides $AB, BC, CD$ and $DA$ of a cyclic quadrilateral $ABCD$. Let $P$ be the point of intersection of $AC$ and $BD$. Prove that the circumradii of triangles $PKL, PLM, PMN$ and $PNK$ are equal to one another.