Olympiad problems

2024 Kyiv City MO Round 1, Problem 2

Let $BL, AD$ be the bisector and the altitude correspondingly of an acute triangle ABC. They intersect at point $T$. It turned out that the altitude $LK$ of $\triangle ALB$ is divided in half by the line $AD$. Prove that $KT \perp BL$. [i]Proposed by Mariia Rozhkova[/i]

2017 Simon Marais Mathematical Competition, B3

Tags: combinatorics , combinatorial geometry , geometry

Each point in the plane with integer coordinates is colored red or blue such that the following two properties hold. For any two red points, the line segment joining them does not contain any blue points. For any two blue points that are distance $2$ apart, the midpoint of the line segment joining them is blue. Prove that if three red points are the vertices of a triangle, then the interior of the triangle does not contain any blue points.

2016 Japan MO Preliminary, 5

Tags: congruent triangles , geometry

Let $ABCD$ be a quadrilateral with $AC=20$, $AD=16$. We take point $P$ on segment $CD$ so that triangle $ABP$ and $ACD$ are congruent. If the area of triangle $APD$ is $28$, find the area of triangle $BCP$. Note that $XY$ expresses the length of segment $XY$.

1998 Spain Mathematical Olympiad, 3

Tags: circumcircle , geometry , trigonometry

Let $ABC$ be a triangle. Points $D$ and $E$ are taken on the line $BC$ such that $AD$ and $AE$ are parallel to the respective tangents to the circumcircle at $C$ and $B$. Prove that \[\frac{BE}{CD}=\left(\frac{AB}{AC}\right)^2 \]

1969 IMO Longlists, 1

Tags: conic , parabola , locus , locus problems , geometry

$(BEL 1)$ A parabola $P_1$ with equation $x^2 - 2py = 0$ and parabola $P_2$ with equation $x^2 + 2py = 0, p > 0$, are given. A line $t$ is tangent to $P_2.$ Find the locus of pole $M$ of the line $t$ with respect to $P_1.$

Estonia Open Senior - geometry, 1996.2.4

Tags: geometry , area , square , trigonometry , circles

The figure shows a square and a circle with a common center $O$, with equal areas of striped shapes. Find the value of $\cos a$. [img]https://2.bp.blogspot.com/-7uwa0H42ELg/XnmsSoPMgcI/AAAAAAAALgk/pHNBqtbsdKgMhcvIRYLm_8JRpOeIYcUeACK4BGAYYCw/s400/96%2Bestonia%2Bopen%2Bs2.4.png[/img]

2009 Junior Balkan Team Selection Tests - Romania, 4

Tags: distance , geometric inequality , geometry

Consider $K$ a polygon in plane, such that the distance between any two vertices is not greater than $1$. Let $X$ and $Y$ be two points inside $K$. Show that there exist a point $Z$, lying on the border of K, such that $XZ + Y Z \le 1$

2007 Bundeswettbewerb Mathematik, 3

Tags: 3d geometry , tetrahedron , geometry , vector

A set $ E$ of points in the 3D space let $ L(E)$ denote the set of all those points which lie on lines composed of two distinct points of $ E.$ Let $ T$ denote the set of all vertices of a regular tetrahedron. Which points are in the set $ L(L(T))?$

2015 Peru IMO TST, 8

Tags: circles , inversion , tangent , geometry

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.

2008 Alexandru Myller, 3

Tags: geometry

Describe all convex, inscriptible polygons which have the property that however we choose three distinct vertexes of of one of them, those vertexes form an isosceles triangle. [i]Gheorghe Iurea[/i]

2016 Tournament Of Towns, 5

Tags: pyramid , geometry , analytic geometry

In convex hexagonal pyramid 11 edges are equal to 1.Find all possible values of 12th edge.

2004 Junior Tuymaada Olympiad, 5

Tags: geometry , geometric transformation , combinatorics

50 knights of King Arthur sat at the Round Table. A glass of white or red wine stood before each of them. It is known that at least one glass of red wine and at least one glass of white wine stood on the table. The king clapped his hands twice. After the first clap every knight with a glass of red wine before him took a glass from his left neighbour. After the second clap every knight with a glass of white wine (and possibly something more) before him gave this glass to the left neughbour of his left neighbour. Prove that some knight was left without wine. [i]Proposed by A. Khrabrov, incorrect translation from Hungarian[/i]

2023 CMWMC, R4

Tags: algebra , combinatorics , geometry

[b]p10.[/b] Square $ABCD$ has side length $n > 1$. Points $E$ and $F$ lie on $\overline{AB}$ and $\overline{BC}$ such that $AE = BF = 1$. Suppose $\overline{DE}$ and $\overline{AF}$ intersect at $X$ and $\frac{AX}{XF} = \frac{11}{111}$ . What is $n$? [b]p11.[/b] Let $x$ be the positive root of $x^2 - 10x - 10 = 0$. Compute $\frac{1}{20}x^4 - 6x^2 - 45$. [b]p12.[/b] Francesca has $7$ identical marbles and $5$ distinctly labeled pots. How many ways are there for her to distribute at least one (but not necessarily all) of the marbles into the pots such that at most two pots are nonempty? PS. You should use hide for answers.

2014 Tuymaada Olympiad, 7

Tags: geometry , analytic geometry , rectangle , geometric transformation , combinatorics

Each of $n$ black squares and $n$ white squares can be obtained by a translation from each other. Every two squares of different colours have a common point. Prove that ther is a point belonging at least to $n$ squares. [i](V. Dolnikov)[/i]

1979 Chisinau City MO, 182

Tags: cube , plane , 3d geometry , pentagon , geometry

Prove that a section of a cube by a plane cannot be a regular pentagon.

2003 Cuba MO, 2

Tags: angle , circles , circumcircle , geometry

Let $A$ be a point outside the circle $\omega$ . The tangents from $A$ touch the circle at $B$ and $C$. Let $P$ be an arbitrary point on extension of $AC$ towards $C$, $Q$ the projection of $C$ onto $PB$ and $E$ the second intersection point of the circumcircle of $ABP$ with the circle $\omega$ . Prove that $\angle PEQ = 2\angle APB$

2007 Czech and Slovak Olympiad III A, 5

Tags: geometry

In an acute-angled triangle $ABC$ ($AC\ne BC$), let $D$ and $E$ be points on $BC$ and $AC$, respectively, such that the points $A,B,D,E$ are concyclic and $AD$ intersects $BE$ at $P$. Knowing that $CP\bot AB$, prove that $P$ is the orthocenter of triangle $ABC$.

2010 Indonesia TST, 1

Tags: geometry , circumcircle , trapezoid

Let $ ABCD$ be a trapezoid such that $ AB \parallel CD$ and assume that there are points $ E$ on the line outside the segment $ BC$ and $ F$ on the segment $ AD$ such that $ \angle DAE \equal{} \angle CBF$. Let $ I,J,K$ respectively be the intersection of line $ EF$ and line $ CD$, the intersection of line $ EF$ and line $ AB$, and the midpoint of segment $ EF$. Prove that $ K$ is on the circumcircle of triangle $ CDJ$ if and only if $ I$ is on the circumcircle of triangle $ ABK$. [i]Utari Wijayanti, Bandung[/i]

1997 Belarusian National Olympiad, 1

Tags: equal segments , cyclic , geometry , perpendicular , pentagon

Different points $A_1,A_2,A_3,A_4,A_5$ lie on a circle so that $A_1A_2 = A_2A_3 = A_3A_4 =A_4A_5$. Let $A_6$ be the diametrically opposite point to $A_2$, and $A_7$ be the intersection of $A_1A_5$ and $A_3A_6$. Prove that the lines $A_1A_6$ and $A_4A_7$ are perpendicular

2024 239 Open Mathematical Olympiad, 8

Tags: geometry , combinatorics

There are $2n$ points on the plane. No three of them lie on the same straight line and no four lie on the same circle. Prove that it is possible to split these points into $n$ pairs and cover each pair of points with a circle containing no other points.

1988 Swedish Mathematical Competition, 2

Tags: combinatorics , geometry , combinatorial geometry

Six ducklings swim on the surface of a pond, which is in the shape of a circle with radius $5$ m. Show that at every moment, two of the ducklings swim on the distance of at most $5$ m from each other.

2006 Mexico National Olympiad, 2

Tags: similar triangles , geometry

Let $ABC$ be a right triangle with a right angle at $A$, such that $AB < AC$. Let $M$ be the midpoint of $BC$ and $D$ the intersection of $AC$ with the perpendicular on $BC$ passing through $M$. Let $E$ be the intersection of the parallel to $AC$ that passes through $M$, with the perpendicular on $BD$ passing through $B$. Show that the triangles $AEM$ and $MCA$ are similar if and only if $\angle ABC = 60^o$.

2007 AMC 12/AHSME, 14

Tags: analytic geometry , geometry

Point $ P$ is inside equilateral $ \triangle ABC$. Points $ Q, R$ and $ S$ are the feet of the perpendiculars from $ P$ to $ \overline{AB}, \overline{BC}$, and $ \overline{CA}$, respectively. Given that $ PQ \equal{} 1, PR \equal{} 2$, and $ PS \equal{} 3$, what is $ AB$? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 3 \sqrt {3}\qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 4 \sqrt {3}\qquad \textbf{(E)}\ 9$

2010 Malaysia National Olympiad, 1

Tags: geometry

A square with side length $2$ cm is placed next to a square with side length $6$ cm, as shown in the diagram. Find the shaded area, in cm$^2$. [img]https://cdn.artofproblemsolving.com/attachments/5/7/ceb4912a6e73ca751113b2b5c92cbfdbb6e0d1.png[/img]

2004 Estonia Team Selection Test, 6

Tags: hexagon , 3d geometry , polyhedron , combinatorial geometry , pentagon , combinatorics , geometry

Call a convex polyhedron a [i]footballoid [/i] if it has the following properties. (1) Any face is either a regular pentagon or a regular hexagon. (2) All neighbours of a pentagonal face are hexagonal (a [i]neighbour [/i] of a face is a face that has a common edge with it). Find all possibilities for the number of pentagonal and hexagonal faces of a footballoid.