Olympiad problems

1993 Tournament Of Towns, (396) 4

A convex $1993$-gon is divided into convex $7$-gons. Prove that there are $3$ neighbouring sides of the $1993$-gon belonging to one such $7$-gon. (A vertex of a $7$-gon may not be positioned on the interior of a side of the $1993$-gon, and two $7$-gons either have no common points, exactly one common vertex or a complete common side.) (A Kanel-Belov)

2000 ITAMO, 5

Tags: grid , minimum , combinatorics , tiling , combinatorial geometry

A man disposes of sufficiently many metal bars of length $2$ and wants to construct a grill of the shape of an $n \times n$ unit net. He is allowed to fold up two bars at an endpoint or to cut a bar into two equal pieces, but two bars may not overlap or intersect. What is the minimum number of pieces he must use?

2024 Sharygin Geometry Olympiad, 8.8

Tags: combinatorial geometry , geometry

Two polygons are cut from the cardboard. Is it possible that for any disposition of these polygons on the plane they have either common inner points or only a finite number of common points on the boundary?

2000 Estonia National Olympiad, 5

Tags: line , combinatorial geometry , combinatorics

At a given plane with $2,000$ lines, all those with an odd number of different points of intersection with intersecting lines. a) Can there be an odd number of red lines if in the plane given there are no parallel lines? b) Can there be an odd number of red lines if none of any 3 given lines intersect at one point?

2025 Sharygin Geometry Olympiad, 13

Tags: geometry , combinatorial geometry

Each two opposite sides of a convex $2n$-gon are parallel. (Two sides are opposite if one passes $n-1$ other sides moving from one side to another along the borderline of the $2n$-gon.) The pair of opposite sides is called regular if there exists a common perpendicular to them such that its endpoints lie on the sides and not on their extensions. Which is the minimal possible number of regular pairs? Proposed by: B.Frenkin

2016 Moldova Team Selection Test, 12

Tags: combinatorics , combinatorial geometry , extremal principle

There are $2015$ distinct circles in a plane, with radius $1$. Prove that you can select $27$ circles, which form a set $C$, which satisfy the following. For two arbitrary circles in $C$, they intersect with each other or For two arbitrary circles in $C$, they don't intersect with each other.

2019 Durer Math Competition Finals, 1

Tags: geometry , combinatorics , combinatorial geometry

Find the number of non-isosceles triangles (up to congruence) with integral side lengths, in which the sum of the two shorter sides is $19$.

1986 Bundeswettbewerb Mathematik, 1

Tags: combinatorics , combinatorial geometry

There are $n$ points on a circle ($n > 1$). Denote them with $P_1,P_2, P_3, ..., P_n$ such that the polyline $P_1P_2P_3... P_n$ does not intersect itself. In how many ways is this possible?

2021 Israel TST, 2

Tags: combinatorial geometry , distance , combinatorics

Let $n>1$ be an integer. Hippo chooses a list of $n$ points in the plane $P_1, \dots, P_n$; some of these points may coincide, but not all of them can be identical. After this, Wombat picks a point from the list $X$ and measures the distances from it to the other $n-1$ points in the list. The average of the resulting $n-1$ numbers will be denoted $m(X)$. Find all values of $n$ for which Hippo can prepare the list in such a way, that for any point $X$ Wombat may pick, he can point to a point $Y$ from the list such that $XY=m(X)$.

1987 Greece National Olympiad, 1

Tags: combinatorial geometry , convex polygon , geometry

It is known that diagonals of a square, as well as a regular pentagon, are all equal. Find the bigeest natural $n$ such that a convex $n$-gon has all it's diagonals equal.

2017 ELMO Shortlist, 3

Tags: geometry , combinatorial geometry

Call the ordered pair of distinct circles $(\omega, \gamma)$ scribable if there exists a triangle with circumcircle $\omega$ and incircle $\gamma$. Prove that among $n$ distinct circles there are at most $(n/2)^2$ scribable pairs. [i]Proposed by Daniel Liu

2007 Moldova Team Selection Test, 4

Tags: ratio , combinatorial geometry , geometry

Consider five points in the plane, no three collinear. The convex hull of this points has area $S$. Prove that there exist three points of them that form a triangle with area at most $\frac{5-\sqrt 5}{10}S$

1985 Czech And Slovak Olympiad IIIA, 3

Tags: vector , inequalities , combinatorics , combinatorial geometry , geometry

If $\overrightarrow{u_1},\overrightarrow{u_2}, ...,\overrightarrow{u_n}$ be vectors in the plane such that the sum of their lengths is at least $1$, then between them we find vectors whose sum is a vector of length at least $\sqrt2/8$. Prove it.

2011 Sharygin Geometry Olympiad, 18

Tags: geometry , combinatorial geometry , line , dissecting plane

On the plane, given are $n$ lines in general position, i.e. any two of them aren’t parallel and any three of them don’t concur. These lines divide the plane into several parts. What is a) the minimal, b) the maximal number of these parts that can be angles?

2016 Denmark MO - Mohr Contest, 2

Tags: combinatorics , coloring , combinatorial geometry

Twenty cubes have been coloured in the following way: There are two red faces opposite each other, two blue faces opposite each other and two green faces opposite each other. The cubes have been glued together as shown in the figure. Two faces that are glued together always have the same colour. The figure shows the colours of some of the faces. Which colours are possible for the face marked with the symbol $\times$? [img]https://cdn.artofproblemsolving.com/attachments/8/2/6127db5bfdce7a749d730fe3626499582f62ba.png[/img]

1974 All Soviet Union Mathematical Olympiad, 193

Tags: vector , combinatorial geometry , combinatorics

Given $n$ vectors of unit length in the plane. The length of their total sum is less than one. Prove that you can rearrange them to provide the property: [i]for every[/i] $k, k\le n$[i], the length of the sum of the first[/i] $k$ [i]vectors is less than[/i] $2$.

1972 Dutch Mathematical Olympiad, 4

Tags: combinatorics , combinatorial geometry

On a circle with radius $1$ the points $A_1, A_2,..., A_n$ lie such that every arc $A_iA_{i+i}$ has length $\frac{2\pi}{n}= a$. Given that there exists a set $V$ consisting of $ k$ of these points ($k < n$), which has the property that each of the arc lengths $a$, $2a$$,...$, $(n- 1)a$ can be obtained in exactly one way be taken as the length of an arc traversed in a positive sense, beginning and ending in a point of $V$. Express $n$ in terms of $k$ and give the set $V$ for the case $n = 7$.

2008 Postal Coaching, 6

Tags: combinatorics , combinatorial geometry , equilateral , point

A set of points in the plane is called [i]free [/i] if no three points of the set are the vertices of an equilateral triangle. Prove that any set of $n$ points in the plane has a free subset of at least $\sqrt{n}$ points

2008 Brazil National Olympiad, 2

Tags: combinatorial geometry , combinatorics

Let $ S$ be a set of $ 6n$ points in a line. Choose randomly $ 4n$ of these points and paint them blue; the other $ 2n$ points are painted green. Prove that there exists a line segment that contains exactly $ 3n$ points from $ S$, $ 2n$ of them blue and $ n$ of them green.

1997 Tournament Of Towns, (536) 1

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

A cube is cut into 99 smaller cubes, exactly 98 of which are unit cubes. Find the volume of the original cube. (V Proizvolov)

1986 Bulgaria National Olympiad, Problem 4

Tags: geometry , combinatorics , combinatorial geometry

Find the smallest integer $n\ge3$ for which there exists an $n$-gon and a point within it such that, if a light bulb is placed at that point, on each side of the polygon there will be a point that is not lightened. Show that for this smallest value of $n$ there always exist two points within the $n$-gon such that the bulbs placed at these points will lighten up the whole perimeter of the $n$-gon.

1992 Poland - Second Round, 1

Tags: geometry , analytic geometry , lattice , combinatorial geometry

Every vertex of a polygon has both integer coordinates; the length of each side of this polygon is a natural number. Prove that the perimeter of the polygon is an even number.

ICMC 8, 6

Tags: combinatorial geometry

A set of points in the plane is called rigid if each point is equidistant from the three (or more) points nearest to it. (a) Does there exist a rigid set of $9$ points? (b) Does there exist a rigid set of $11$ points?

1966 IMO Shortlist, 49

Tags: geometry , reflection , combinatorial geometry

Two mirror walls are placed to form an angle of measure $\alpha$. There is a candle inside the angle. How many reflections of the candle can an observer see?

1997 IMO Shortlist, 3

Tags: vector , combinatorics , combinatorial geometry , extremal combinatorics , geometry

For each finite set $ U$ of nonzero vectors in the plane we define $ l(U)$ to be the length of the vector that is the sum of all vectors in $ U.$ Given a finite set $ V$ of nonzero vectors in the plane, a subset $ B$ of $ V$ is said to be maximal if $ l(B)$ is greater than or equal to $ l(A)$ for each nonempty subset $ A$ of $ V.$ (a) Construct sets of 4 and 5 vectors that have 8 and 10 maximal subsets respectively. (b) Show that, for any set $ V$ consisting of $ n \geq 1$ vectors the number of maximal subsets is less than or equal to $ 2n.$