Olympiad problems

2020 Candian MO, 3#

Tags: geometry , circumcircle , ratio , Really old , geometry unsolved , algebra

okay this one is from Prof. Mircea Lascu from Zalau, Romaniaand Prof. V. Cartoaje from Ploiesti, Romania. It goes like this: given being a triangle ABC for every point M inside we construct the points A[size=67]M[/size], B[size=67]M[/size], C[size=67]M[/size] on the circumcircle of the triangle ABC such that A, A[size=67]M[/size], M are collinear and so on. Find the locus of these points M for which the area of the triangle A[size=67]M[/size] B[size=67]M[/size] C[size=67]M[/size] is bigger than the area of the triangle ABC.

2011 India National Olympiad, 5

Tags: geometry , cyclic quadrilateral , geometry unsolved

Let $ABCD$ be a cyclic quadrilateral inscribed in a circle $\Gamma.$ Let $E,F,G,H$ be the midpoints of arcs $AB,BC,CD,AD$ of $\Gamma,$ respectively. Suppose that $AC\cdot BD=EG\cdot FH.$ Show that $AC,BD,EG,FH$ are all concurrent.

2004 Czech-Polish-Slovak Match, 3

Tags: geometry , cyclic quadrilateral , geometry unsolved

A point P in the interior of a cyclic quadrilateral ABCD satisfies ∠BPC = ∠BAP + ∠PDC. Denote by E, F and G the feet of the perpendiculars from P to the lines AB, AD and DC, respectively. Show that the triangles FEG and PBC are similar.

2009 Nordic, 1

Tags: ratio , geometry , geometry unsolved

A point $P$ is chosen in an arbitrary triangle. Three lines are drawn through $P$ which are parallel to the sides of the triangle. The lines divide the triangle into three smaller triangles and three parallelograms. Let $f$ be the ratio between the total area of the three smaller triangles and the area of the given triangle. Prove that $f\ge\frac{1}{3}$ and determine those points $P$ for which $f =\frac{1}{3}$ .

2007 Romania National Olympiad, 2

Tags: geometry , geometry unsolved

Let $ABC$ be an acute angled triangle and point $M$ chosen differently from $A,B,C$. Prove that $M$ is the orthocenter of triangle $ABC$ if and only if \[\frac{BC}{MA}\vec{MA}+\frac{CA}{MB}\vec{MB}+\frac{AB}{MC}\vec{MC}= \vec{0}\]

1976 IMO Longlists, 7

Tags: vector , geometry unsolved , geometry

Let $P$ be a fixed point and $T$ a given triangle that contains the point $P$. Translate the triangle $T$ by a given vector $\bold{v}$ and denote by $T'$ this new triangle. Let $r, R$, respectively, be the radii of the smallest disks centered at $P$ that contain the triangles $T , T'$, respectively. Prove that $r + |\bold{v}| \leq 3R$ and find an example to show that equality can occur.

2001 Tournament Of Towns, 4

Tags: induction , geometry unsolved , geometry

Several non-intersecting diagonals divide a convex polygon into triangles. At each vertex of the polygon the number of triangles adjacent to it is written. Is it possible to reconstruct all the diagonals using these numbers if the diagonals are erased?

1995 China Team Selection Test, 2

Tags: geometry , geometric transformation , geometry unsolved , Tangents , Locus problems , China , TST

Given a fixed acute angle $\theta$ and a pair of internally tangent circles, let the line $l$ which passes through the point of tangency, $A$, cut the larger circle again at $B$ ($l$ does not pass through the centers of the circles). Let $M$ be a point on the major arc $AB$ of the larger circle, $N$ the point where $AM$ intersects the smaller circle, and $P$ the point on ray $MB$ such that $\angle MPN = \theta$. Find the locus of $P$ as $M$ moves on major arc $AB$ of the larger circle.

2012 Indonesia TST, 3

Tags: geometry , rectangle , geometry unsolved

Suppose $l(M, XYZ)$ is a Simson line of the triangle $XYZ$ that passes through $M$. Suppose $ABCDEF$ is a cyclic hexagon such that $l(A, BDF), l(B, ACE), l(D, ABF), l(E, ABC)$ intersect at a single point. Prove that $CDEF$ is a rectangle. [color=blue]Should the first sentence read: Suppose $l(M, XYZ)$ is a Simson line of the triangle $XYZ$ [u]with respect to[/u] $M$. ? Since it appears weird that a Simson line that passes a point is to be constructed. However, this is Unsolved after all, so I'm not sure.[/color]

2002 Indonesia MO, 7

Tags: geometry , rhombus , geometry unsolved

Let $ABCD$ be a rhombus where $\angle DAB = 60^\circ$, and $P$ be the intersection between $AC$ and $BD$. Let $Q,R,S$ be three points on the boundary of $ABCD$ such that $PQRS$ is a rhombus. Prove that exactly one of $Q,R,S$ lies on one of $A,B,C,D$.

2006 India Regional Mathematical Olympiad, 5

Tags: geometry , inradius , geometry unsolved , nice problem

Let $ ABCD$ be a quadrilateral in which $ AB$ is parallel to $ CD$ and perpendicular to $ AD; AB \equal{} 3CD;$ and the area of the quadrilateral is $ 4$. if a circle can be drawn touching all the four sides of the quadrilateral, find its radius.

2005 Irish Math Olympiad, 1

Tags: inequalities , geometry , incenter , geometry unsolved

Let $ X$ be a point on the side $ AB$ of a triangle $ ABC$, different from $ A$ and $ B$. Let $ P$ and $ Q$ be the incenters of the triangles $ ACX$ and $ BCX$ respectively, and let $ M$ be the midpoint of $ PQ$. Prove that: $ MC>MX$.

2010 Czech-Polish-Slovak Match, 3

Tags: geometry , parallelogram , geometric transformation , reflection , inequalities , triangle inequality , geometry unsolved

Let $ABCD$ be a convex quadrilateral for which \[ AB+CD=\sqrt{2}\cdot AC\qquad\text{and}\qquad BC+DA=\sqrt{2}\cdot BD.\] Prove that $ABCD$ is a parallelogram.

1988 IMO Longlists, 59

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

In $3$-dimensional space there is given a point $O$ and a finite set $A$ of segments with the sum of lengths equal to $1988$. Prove that there exists a plane disjoint from $A$ such that the distance from it to $O$ does not exceed $574$.

2000 Baltic Way, 4

Tags: geometry , parallelogram , geometry unsolved

Given a triangle $ ABC$ with $ \angle A \equal{} 120^{\circ}$. The points $ K$ and $ L$ lie on the sides $ AB$ and $ AC$, respectively. Let $ BKP$ and $ CLQ$ be equilateral triangles constructed outside the triangle $ ABC$. Prove that $ PQ \ge\frac{\sqrt 3}{2}\left(AB \plus{} AC\right)$.

2006 China Girls Math Olympiad, 2

Tags: geometry , circumcircle , geometry unsolved

Let $O$ be the intersection of the diagonals of convex quadrilateral $ABCD$. The circumcircles of $\triangle{OAD}$ and $\triangle{OBC}$ meet at $O$ and $M$. Line $OM$ meets the circumcircles of $\triangle{OAB}$ and $\triangle{OCD}$ at $T$ and $S$ respectively. Prove that $M$ is the midpoint of $ST$.

2014 Miklós Schweitzer, 10

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

To each vertex of a given triangulation of the two-dimensional sphere, we assign a convex subset of the plane. Assume that the three convex sets corresponding to the three vertices of any two-dimensional face of the triangulation have at least one point in common. Show that there exist four vertices such that the corresponding convex sets have at least one point in common.

1972 IMO Longlists, 38

Tags: geometry , rectangle , geometry unsolved

Congruent rectangles with sides $m(cm)$ and $n(cm)$ are given ($m, n$ positive integers). Characterize the rectangles that can be constructed from these rectangles (in the fashion of a jigsaw puzzle). (The number of rectangles is unbounded.)

1979 IMO Longlists, 75

Tags: geometry unsolved , geometry

Given an equilateral triangle $ABC$, let $M$ be an arbitrary point in space. $(\text{a})$ Prove that one can construct a triangle from the segments $MA, MB, MC$. $(\text{b})$ Suppose that $P$ and $Q$ are two points symmetric with respect to the center $O$ of $ABC$. Prove that the two triangles constructed from the segments $PA,PB,PC$ and $QA,QB,QC$ are of equal area.

2007 Pan African, 2

Tags: geometry unsolved , geometry

Let $A$, $B$ and $C$ be three fixed points, not on the same line. Consider all triangles $AB'C'$ where $B'$ moves on a given straight line (not containing $A$), and $C'$ is determined such that $\angle B'=\angle B$ and $\angle C'=\angle C$. Find the locus of $C'$.

1984 Vietnam National Olympiad, 3

Tags: geometry unsolved , geometry

A square $ABCD$ of side length $2a$ is given on a plane $\Pi$. Let $S$ be a point on the ray $Ax$ perpendicular to $\Pi$ such that $AS = 2a.$ $(a)$ Let $M \in BC$ and $N \in CD$ be two variable points. $i$. Find the positions of $M,N$ such that $BM + DN \ge \frac{3}{2}$, planes $SAM$ and $SMN$ are perpendicular and $BM \cdot DN$ is minimum. $ii$. Find $M$ and $N$ such that $\angle MAN = 45^{\circ}$ and the volume of $SAMN$ attains an extremum value. Find these values. $(b)$ Let $Q$ be a point such that $\angle AQB = \angle AQD = 90^{\circ}$. The line $DQ$ intersects the plane $\pi$ through $AB$ perpendicular to $\Pi$ at $Q'$. $i$. Find the locus of $Q'$. $ii$. Let $K$ be the locus of points $Q$ and let $CQ$ meet $K$ again at $R$. Let $DR$ meets $\Pi$ at $R'$. Prove that $sin^2 \angle Q'DB + sin^2 \angle R'DB$ is independent of $Q$.

2014 Greece National Olympiad, 4

Tags: projective geometry , geometry , cyclic quadrilateral , geometry unsolved

We are given a circle $c(O,R)$ and two points $A,B$ so that $R<AB<2R$.The circle $c_1 (A,r)$ ($0<r<R$) crosses the circle $c$ at C,D ($C$ belongs to the short arc $AB$).From $B$ we consider the tangent lines $BE,BF$ to the circle $c_1$ ,in such way that $E$ lays out of the circle $c$.If $M\equiv EC\cap DF$ show that the quadrilateral $BCFM$ is cyclic.

2011 Postal Coaching, 1

Tags: geometry , circumcircle , incenter , geometry unsolved

Let $ABC$ be a triangle in which $\angle BAC = 60^{\circ}$ . Let $P$ (similarly $Q$) be the point of intersection of the bisector of $\angle ABC$(similarly of $\angle ACB$) and the side $AC$(similarly $AB$). Let $r_1$ and $r_2$ be the in-radii of the triangles $ABC$ and $AP Q$, respectively. Determine the circum-radius of $APQ$ in terms of $r_1$ and $r_2$.

2003 Italy TST, 2

Tags: geometry , circumcircle , geometry unsolved

Let $B\not= A$ be a point on the tangent to circle $S_1$ through the point $A$ on the circle. A point $C$ outside the circle is chosen so that segment $AC$ intersects the circle in two distinct points. Let $S_2$ be the circle tangent to $AC$ at $C$ and to $S_1$ at some point $D$, where $D$ and $B$ are on the opposite sides of the line $AC$. Let $O$ be the circumcentre of triangle $BCD$. Show that $O$ lies on the circumcircle of triangle $ABC$.

1983 USAMO, 4

Tags: AMC , USA(J)MO , USAMO , geometry , 3D geometry , tetrahedron , geometry unsolved

Six segments $S_1, S_2, S_3, S_4, S_5,$ and $S_6$ are given in a plane. These are congruent to the edges $AB, AC, AD, BC, BD,$ and $CD$, respectively, of a tetrahedron $ABCD$. Show how to construct a segment congruent to the altitude of the tetrahedron from vertex $A$ with straight-edge and compasses.