Olympiad problems

2018 Oral Moscow Geometry Olympiad, 6

Cut each of the equilateral triangles with sides $2$ and $3$ into three parts and construct an equilateral triangle from all received parts.

2000 Tournament Of Towns, 2

Tags: geometry , 3d geometry , combinatorial geometry , combinatorics , regular polygon , cube

What is the largest integer $n$ such that one can find $n$ points on the surface of a cube, not all lying on one face and being the vertices of a regular $n$-gon? (A Shapovalov)

2018 PUMaC Combinatorics A, 8

Tags: combinatorics , combinatorial geometry

Let $S_5$ be the set of permutations of $\{1,2,3,4,5\}$, and let $C$ be the convex hull of the set $$\{(\sigma(1),\sigma(2),\ldots,\sigma(5))\,|\,\sigma\in S_5\}.$$ Then $C$ is a polyhedron. What is the total number of $2$-dimensional faces of $C$?

1997 Argentina National Olympiad, 1

Tags: combinatorics , combinatorial geometry , geometry

Let $s$ and $t$ be two parallel lines. We have marked $k$ points on line $s$ and $n$ points on line $t$ ($k\geq n$). If it is known that the total number of triangles that have their three vertices at marked points is $220$, find all possible values of $k$ and $n$.

2016 Saint Petersburg Mathematical Olympiad, 4

Tags: point , combinatorial geometry , combinatorics , speed , circles

$N> 4$ points move around the circle, each with a constant speed. For Any four of them have a moment in time when they all meet. Prove that is the moment when all the points meet.

2001 Estonia Team Selection Test, 1

Tags: geometry , rectangle , combinatorial geometry , combinatorics

Consider on the coordinate plane all rectangles whose (i) vertices have integer coordinates; (ii) edges are parallel to coordinate axes; (iii) area is $2^k$, where $k = 0,1,2....$ Is it possible to color all points with integer coordinates in two colors so that no such rectangle has all its vertices of the same color?

2019 Czech-Polish-Slovak Junior Match, 5

Tags: geometry , geometric transformation , rotation , combinatorial geometry , combinatorics

Let $A_1A_2 ...A_{360}$ be a regular $360$-gon with centre $S$. For each of the triangles $A_1A_{50}A_{68}$ and $A_1A_{50}A_{69}$ determine, whether its images under some $120$ rotations with centre $S$ can have (as triangles) all the $360$ points $A_1, A_2, ..., A_{360}$ as vertices.

1987 Tournament Of Towns, (140) 5

Tags: coloring , combinatorial geometry , combinatorics , cube , rectangle

A certain number of cubes are painted in six colours, each cube having six faces of different colours (the colours in different cubes may be arranged differently) . The cubes are placed on a table so as to form a rectangle. We are allowed to take out any column of cubes, rotate it (as a whole) along its long axis and replace it in the rectangle. A similar operation with rows is also allowed. Can we always make the rectangle monochromatic (i.e. such that the top faces of all the cubes are the same colour) by means of such operations? ( D. Fomin , Leningrad)

1973 Polish MO Finals, 3

Tags: geometry , 3d geometry , combinatorial geometry , polyhedron , parallelepiped

A polyhedron $W$ has the following properties: (i) It possesses a center of symmetry. (ii) The section of $W$ by a plane passing through the center of symmetry and one of its edges is always a parallelogram. (iii) There is a vertex of $W$ at which exactly three edges meet. Prove that $W$ is a parallelepiped.

2018 BAMO, E/3

Tags: combinatorial geometry , combinatorics

Suppose that $2002$ numbers, each equal to $1$ or $-1$, are written around a circle. For every two adjacent numbers, their product is taken; it turns out that the sum of all $2002$ such products is negative. Prove that the sum of the original numbers has absolute value less than or equal to $1000$. (The absolute value of $x$ is usually denoted by $|x|$. It is equal to $x$ if $x \ge 0$, and to $-x$ if $x < 0$. For example, $|6| = 6, |0| = 0$, and $|-7| = 7$.)

2012 IMAR Test, 4

Tags: combinatorial geometry , coloring , geometry

Design a planar finite non-empty set $S$ satisfying the following two conditions: (a) every line meets $S$ in at most four points; and (b) every $2$-colouring of $S$ - that is, each point of $S$ is coloured one of two colours - yields (at least) three monochromatic collinear points.

KoMaL A Problems 2021/2022, A. 814

Tags: geometry , combinatorial geometry

We are given $666$ points on the plane such that they cannot be covered by $10$ lines. Show that we can choose $66$ out of these points such that they can not be covered by $10$ lines.

2004 Germany Team Selection Test, 2

Tags: geometry , combinatorial geometry , polygon , extremal combinatorics , right angle , angle

Let $n \geq 5$ be a given integer. Determine the greatest integer $k$ for which there exists a polygon with $n$ vertices (convex or not, with non-selfintersecting boundary) having $k$ internal right angles. [i]Proposed by Juozas Juvencijus Macys, Lithuania[/i]

1965 Kurschak Competition, 2

Tags: combinatorics , combinatorial geometry

$D$ is a closed disk radius $R$. Show that among any $8$ points of $D$ one can always find two whose distance apart is less than $R$.

2018-IMOC, G1

Tags: point , geometry , combinatorial geometry , polygon

Given an integer $n \ge 3$. Find the largest positive integer $k $ with the following property: For $n$ points in general position, there exists $k$ ways to draw a non-intersecting polygon with those $n$ points as it’s vertices. [hide=Different wording]Given $n$, find the maximum $k$ so that for every general position of $n$ points , there are at least $k$ ways of connecting the points to form a polygon.[/hide]

2007 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: combinatorial geometry , number theory , geometry , rectangle

Determine the values of $n \in N$ such that a square of side $n$ can be split into a square of side $1$ and five rectangles whose side measures are $10$ distinct natural numbers and all greater than $1$.

1970 Polish MO Finals, 4

Tags: geometry , rectangle , combinatorial geometry

In the plane are given two mutually perpendicular lines and $n$ rectangles with sides parallel to the two lines. Show that if every two rectangles have a common point, then all the rectangles have a common point.

2016 Regional Olympiad of Mexico West, 6

Tags: coloring , combinatorics , combinatorial geometry

The vertices of a regular polygon with $2016$ sides are colored gold or silver. Prove that there are at least $512$ different isosceles triangles whose vertices have the same color.

1978 All Soviet Union Mathematical Olympiad, 259

Tags: square , combinatorial geometry , combinatorics

Prove that there exists such a number $A$ that you can inscribe $1978$ different size squares in the plot of the function $y = A sin(x)$. (The square is inscribed if all its vertices belong to the plot.)

2009 BAMO, 1

Tags: combinatorial geometry , combinatorics

A square grid of $16$ dots (see the figure) contains the corners of nine $1\times1$ squares, four $2\times 2$ squares, and one $3\times3$ square, for a total of $14$ squares whose sides are parallel to the sides of the grid. What is the smallest possible number of dots you can remove so that, after removing those dots, each of the $14$ squares is missing at least one corner? Justify your answer by showing both that the number of dots you claim is sufficient and by explaining why no smaller number of dots will work. [img]https://cdn.artofproblemsolving.com/attachments/0/9/bf091a769dbec40eceb655f5588f843d4941d6.png[/img]

2015 QEDMO 14th, 5

Tags: combinatorics , combinatorial geometry

Let $D$ be a regular dodecagon in the plane. How many squares are there in the plane at least two vertices in common with the vertices of $D$?

2021 Final Mathematical Cup, 4

Tags: combinatorics , combinatorial geometry

A number of $n$ lamps ($n\ge 3$) are put at $n$ vertices of a regular $n$-gon. Initially, all the lamps are off. In each step. Lisa will choose three lamps that are located at three vertices of an isosceles triangle and change their states (from off to on and vice versa). Her aim is to turn on all the lamps. At least how many steps are required to do so?

2003 IMO Shortlist, 3

Tags: geometry , combinatorial geometry , polygon , extremal combinatorics , right angle , angle

Let $n \geq 5$ be a given integer. Determine the greatest integer $k$ for which there exists a polygon with $n$ vertices (convex or not, with non-selfintersecting boundary) having $k$ internal right angles. [i]Proposed by Juozas Juvencijus Macys, Lithuania[/i]

2020 Costa Rica - Final Round, 6

Tags: combinatorics , combinatorial geometry , geometry

$10$ persons sit around a circular table and on the table there are $22$ vases. Two persons can see each other if and only if there are no vases aligned with them. Prove that there are at least two people who can see each other.

1945 Moscow Mathematical Olympiad, 100

Tags: polygon , minimum , combinatorial geometry , geometry

Suppose we have two identical cardboard polygons. We placed one polygon upon the other one and aligned. Then we pierced polygons with a pin at a point. Then we turned one of the polygons around this pin by $25^o 30'$. It turned out that the polygons coincided (aligned again). What is the minimal possible number of sides of the polygons?