Olympiad problems

1972 Poland - Second Round, 4

A cube with edge length $ n $ is divided into $ n^3 $ unit cubes by planes parallel to its faces. How many pairs of such unit cubes exist that have no more than two vertices in common?

1986 Tournament Of Towns, (131) 7

Tags: combinatorics , combinatorial geometry , arc , angle , circles , point

On the circumference of a circle are $21$ points. Prove that among the arcs which join any two of these points, at least $100$ of them must subtend an angle at the centre of the circle not exceeding $120^o$ . ( A . F . Sidorenko)

2016 Hanoi Open Mathematics Competitions, 7

Tags: combinatorics , combinatorial geometry , grid

Nine points form a grid of size $3\times 3$. How many triangles are there with $3$ vertices at these points?

1970 Polish MO Finals, 4

Tags: rectangle , combinatorial geometry , 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.

1997 Chile National Olympiad, 6

Tags: combinatorial geometry , combinatorics

For each set $C$ of points in space, we designate by $P_C$ the set of planes containing at least three points of $C$. $\bullet$ Prove that there exists $C$ such that $\phi (P_C) = 1997$, where $\phi$ corresponds to the cardinality. $\bullet$ Determine the least number of points that $C$ must have so that the previous property can be fulfilled.

2019 Iran Team Selection Test, 5

Tags: combinatorial geometry , combinatorics , perimeter

Let $P$ be a simple polygon completely in $C$, a circle with radius $1$, such that $P$ does not pass through the center of $C$. The perimeter of $P$ is $36$. Prove that there is a radius of $C$ that intersects $P$ at least $6$ times, or there is a circle which is concentric with $C$ and have at least $6$ common points with $P$. [i]Proposed by Seyed Reza Hosseini[/i]

1990 Tournament Of Towns, (245) 3

Tags: combinatorics , combinatorial geometry , coloring

Is it possible to put together $27$ equal cubes, $9$ red, $9$ blue and $9$ white, so as to obtain a big cube in which each row (parallel to an arbitrary edge of the cube) contains three cubes with exactly two different colours? (S. Fomin, Leningrad)

1999 All-Russian Olympiad Regional Round, 8.8

Tags: combinatorics , combinatorial geometry , 3d geometry , geometry

An open chain was made from $54$ identical single cardboard squares, connecting them hingedly at the vertices. Any square (except for the extreme ones) is connected to its neighbors by two opposite vertices. Is it possible to completely cover a $3\times 3 \times3$ surface with this chain of squares?

2025 Kyiv City MO Round 1, Problem 3

Tags: geometry , combinatorial geometry

What's the smallest positive integer $ n > 3 $, for which there does [b]not[/b] exist a (not necessarily convex) $ n $-gon such that all its diagonals have equal lengths? A diagonal of any polygon is defined as a segment connecting any two non-adjacent vertices of the polygon. [i]Proposed by Anton Trygub[/i]

2021 China Team Selection Test, 1

Tags: combinatorial geometry , combinatorics , graph theory

Given positive integer $ n \ge 5 $ and a convex polygon $P$, namely $ A_1A_2...A_n $. No diagonals of $P$ are concurrent. Proof that it is possible to choose a point inside every quadrilateral $ A_iA_jA_kA_l (1\le i<j<k<l\le n) $ not on diagonals of $P$, such that the $ \tbinom{n}{4} $ points chosen are distinct, and any segment connecting these points intersect with some diagonal of P.

1999 VJIMC, Problem 1

Tags: geometry , combinatorics , combinatorial geometry , 3d geometry

Find the minimal $k$ such that every set of $k$ different lines in $\mathbb R^3$ contains either $3$ mutually parallel lines or $3$ mutually intersecting lines or $3$ mutually skew lines.

2011 District Olympiad, 1

Tags: combinatorics , geometry , combinatorial geometry

In a square of side length $60$, $121$ distinct points are given. Show that among them there exists three points which are vertices of a triangle with an area not exceeding $30$.

2016 IFYM, Sozopol, 1

Tags: combinatorial geometry , combinatorics

The numbers from 1 to $n$ are arranged in some way on a circle. What’s the smallest value of $n$, for which no matter how the numbers are arranged there exist ten consecutively increasing numbers $A_1<A_2<A_3…<A_{10}$ such that $A_1 A_2…A_{10}$ is a convex decagon?

2002 Junior Balkan Team Selection Tests - Moldova, 2

Tags: combinatorics , combinatorial geometry , line , point

$64$ distinct points are positioned in the plane so that they determine exactly $2003$ different lines. Prove that among the $64$ points there are at least $4$ collinear points.

1991 Czech And Slovak Olympiad IIIA, 2

Tags: geometry , combinatorics , combinatorial geometry

A museum has the shape of a (not necessarily convex) 3$n$-gon. Prove that $n$ custodians can be positioned so as to control all of the museum’s space.

1995 Tournament Of Towns, (459) 4

Tags: geometry , combinatorics , combinatorial geometry , lattice points

Some points with integer coordinates in the plane are marked. It is known that no four of them lie on a circle. Show that there exists a circle of radius 1995 without any marked points inside. (AV Shapovelov)

1972 Dutch Mathematical Olympiad, 1

Tags: geometry , combinatorics , combinatorial geometry , equilateral

Prove that for every $n \in N$, $n > 6$, every equilateral triangle can be divided into $n$ pieces, which are also equilateral triangles.

2007 JBMO Shortlist, 2

Tags: combinatorics , combinatorial geometry

Given are $50$ points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least $130$ scalene triangles with vertices of that color.

2021 Czech-Polish-Slovak Junior Match, 3

Tags: combinatorial geometry , combinatorics

A [i]cross [/i] is the figure composed of $6$ unit squares shown below (and any figure made of it by rotation). [img]https://cdn.artofproblemsolving.com/attachments/6/0/6d4e0579d2e4c4fa67fd1219837576189ec9cb.png[/img] Find the greatest number of crosses that can be cut from a $6 \times 11$ divided sheet of paper into unit squares (in such a way that each cross consists of six such squares).

1996 Estonia National Olympiad, 5

Tags: combinatorial geometry , combinatorics , tetrahedron

Suppose that $n$ teterahedra are given in space such that any two of them have at least two common vertices, but any three have at most one common vertex. Find the greatest possible $n$.

1984 Bundeswettbewerb Mathematik, 2

Tags: geometry , combinatorial geometry , regular polygon

Given is a regular $n$-gon with circumradius $1$. $L$ is the set of (different) lengths of all connecting segments of its endpoints. What is the sum of the squares of the elements of $L$?

2013 Portugal MO, 6

Tags: combinatorics , regular , equal angles , combinatorial geometry , geometry

In each side of a regular polygon with $n$ sides, we choose a point different from the vertices and we obtain a new polygon of $n$ sides. For which values of $n$ can we obtain a polygon such that the internal angles are all equal but the polygon isn't regular?

2001 All-Russian Olympiad Regional Round, 10.4

Tags: geometry , combinatorics , combinatorial geometry

Three families of parallel lines are drawn,$10$ lines each, are drawn. What is the greatest number of triangles they can cut from plane?

Cono Sur Shortlist - geometry, 2009.G1.6

Tags: geometry , combinatorial geometry , rectangle

Sebastian has a certain number of rectangles with areas that sum up to 3 and with side lengths all less than or equal to $1$. Demonstrate that with each of these rectangles it is possible to cover a square with side $1$ in such a way that the sides of the rectangles are parallel to the sides of the square. [b]Note:[/b] The rectangles can overlap and they can protrude over the sides of the square.

2014 Greece National Olympiad, 3

Tags: euler , modular arithmetic , combinatorial geometry , combinatorics , discrete intermediate value theorem

For even positive integer $n$ we put all numbers $1,2,...,n^2$ into the squares of an $n\times n$ chessboard (each number appears once and only once). Let $S_1$ be the sum of the numbers put in the black squares and $S_2$ be the sum of the numbers put in the white squares. Find all $n$ such that we can achieve $\frac{S_1}{S_2}=\frac{39}{64}.$