This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 1704

2010 Belarus Team Selection Test, 6.3
Tags: combinatorics , combinatorial geometry , tiling , tiles
A $50 \times 50$ square board is tiled by the tetrominoes of the following three types: [img]https://cdn.artofproblemsolving.com/attachments/2/9/62c0bce6356ea3edd8a2ebfe0269559b7527f1.png[/img] Find the greatest and the smallest possible number of $L$ -shaped tetrominoes In the tiling. (Folklore)

1990 Bundeswettbewerb Mathematik, 2
Tags: combinatorial geometry , combinatorics
Let $A(n)$ be the least possible number of distinct points in the plane with the following property: For every $k = 1,2,...,n$ there is a line containing precisely $k$ of these points. Show that $A(n) =\left[\frac{n+1}{2}\right] \left[\frac{n+2}{2}\right]$

1984 Czech And Slovak Olympiad IIIA, 5
Tags: geometry , combinatorial geometry , 3d geometry , polyhedron , combinatorics
Find all natural numbers $n$ for which there exists a convex polyhedron with $n$ edges, with exactly one vertex having four edges and all other vertices having $3$ edges.

2020 Silk Road, 4
Tags: combinatorial geometry , convex hull , combinatorics
Prove that for any natural number $ m $ there exists a natural number $ n $ such that any $ n $ distinct points on the plane can be partitioned into $ m $ non-empty sets whose [i]convex hulls[/i] have a common point. The [i] convex hull [/i] of a finite set $ X $ of points on the plane is the set of points lying inside or on the boundary of at least one convex polygon with vertices in $ X $, including degenerate ones, that is, the segment and the point are considered convex polygons. No three vertices of a convex polygon are collinear. The polygon contains its border.

1986 Bulgaria National Olympiad, Problem 4
Tags: combinatorial geometry , geometry , combinatorics
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.

1997 Croatia National Olympiad, Problem 4
Tags: combinatorial geometry , geometry , rectangle , combinatorics
An infinite sheet of paper is divided into equal squares, some of which are colored red. In each $2\times3$ rectangle, there are exactly two red squares. Now consider an arbitrary $9\times11$ rectangle. How many red squares does it contain? (The sides of all considered rectangles go along the grid lines.)

1995 Tournament Of Towns, (459) 4
Tags: geometry , combinatorial geometry , lattice points , combinatorics
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)

1999 IMO Shortlist, 2
Tags: combinatorial geometry , circles , geometry , point set
A circle is called a [b]separator[/b] for a set of five points in a plane if it passes through three of these points, it contains a fourth point inside and the fifth point is outside the circle. Prove that every set of five points such that no three are collinear and no four are concyclic has exactly four separators.

2004 Estonia Team Selection Test, 3
Tags: combinatorics , combinatorial geometry , regular polygon
For which natural number $n$ is it possible to draw $n$ line segments between vertices of a regular $2n$-gon so that every vertex is an endpoint for exactly one segment and these segments have pairwise different lengths?

1974 All Soviet Union Mathematical Olympiad, 193
Tags: combinatorial geometry , vector , 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] $2$.

2018 Bundeswettbewerb Mathematik, 4
Tags: combinatorial geometry , combinatorics
Determine alle positive integers $n>1$ with the following property: For each colouring of the lattice points in the plane with $n$ colours, there are three lattice points of the same colour forming an isosceles right triangle with legs parallel to the coordinate axes.

2007 Sharygin Geometry Olympiad, 6
Tags: combinatorial geometry , circles , geometry
Given are two concentric circles $\Omega$ and $\omega$. Each of the circles $b_1$ and $b_2$ is externally tangent to $\omega$ and internally tangent to $\Omega$, and $\omega$ each of the circles $c_1$ and $c_2$ is internally tangent to both $\Omega$ and $\omega$. Mark each point where one of the circles $b_1, b_2$ intersects one of the circles $c_1$ and $c_2$. Prove that there exist two circles distinct from $b_1, b_2, c_1, c_2$ which contain all $8$ marked points. (Some of these new circles may appear to be lines.)

1986 Tournament Of Towns, (121) 3
Tags: combinatorial geometry , game , combinatorics , game strategy
A game has two players. In the game there is a rectangular chocolate bar, with $60$ pieces, arranged in a $6 \times 1 0$ formation , which can be broken only along the lines dividing the pieces. The first player breaks the bar along one line, discarding one section . The second player then breaks the remaining section, discarding one section. The first player repeats this process with the remaining section , and so on. The game is won by the player who leaves a single piece. In a perfect game which player wins? {S. Fomin , Leningrad)

1998 All-Russian Olympiad Regional Round, 8.4
Tags: combinatorics , combinatorial geometry , geometry
A set of $n\ge 9$ points is given on the plane. For any 9 it points can be selected from all circles so that all these points end up on selected circles. Prove that all n points lie on two circles

2018 Hanoi Open Mathematics Competitions, 9
Tags: combinatorial geometry , area , geometric inequality , geometry
There are three polygons and the area of each one is $3$. They are drawn inside a square of area $6$. Find the greatest value of $m$ such that among those three polygons, we can always find two polygons so that the area of their overlap is not less than $m$.

2024 India IMOTC, 24
Tags: geometry , combinatorial geometry , combinatorics
There are $n > 1$ distinct points marked in the plane. Prove that there exists a set of circles $\mathcal C$ such that [color=#FFFFFF]___[/color]$\bullet$ Each circle in $\mathcal C$ has unit radius. [color=#FFFFFF]___[/color]$\bullet$ Every marked point lies in the (strict) interior of some circle in $\mathcal C$. [color=#FFFFFF]___[/color]$\bullet$ There are less than $0.3n$ pairs of circles in $\mathcal C$ that intersect in exactly $2$ points. [i]Note: Weaker results with $\it{0.3n}$ replaced by $\it{cn}$ may be awarded points depending on the value of the constant $\it{c > 0.3}$.[/i] [i]Proposed by Siddharth Choppara, Archit Manas, Ananda Bhaduri, Manu Param[/i]

2007 Hanoi Open Mathematics Competitions, 7
Tags: combinatorial geometry , equilateral , geometry
Nine points, no three of which lie on the same straight line, are located inside an equilateral triangle of side $4$. Prove that some three of these points are vertices of a triangle whose area is not greater than $\sqrt3$.

2020 Puerto Rico Team Selection Test, 4
Tags: combinatorial geometry , combinatorics , rectangle
Determine all integers $m$, for which it is possible to dissect the square $m\times m$ into five rectangles, with the side lengths being the integers $1, 2, … ,10$ in some order.

1975 All Soviet Union Mathematical Olympiad, 211
Tags: combinatorial geometry , combinatorics , points in space
Given a finite set of polygons in the plane. Every two of them have a common point. Prove that there exists a straight line, that crosses all the polygons.

1972 Swedish Mathematical Competition, 2
Tags: combinatorics , combinatorial geometry
A rectangular grid of streets has $m$ north-south streets and $n$ east-west streets. For which $m, n > 1$ is it possible to start at an intersection and drive through each of the other intersections just once before returning to the start?

2012 JBMO ShortLists, 3
Tags: combinatorics , combinatorial geometry , geometry
In a circle of diameter $1$ consider $65$ points, no three of them collinear. Prove that there exist three among these points which are the vertices of a triangle with area less than or equal to $\frac{1}{72}$.

2012 Rioplatense Mathematical Olympiad, Level 3, 3
Tags: angle bisector , combinatorial geometry , parallel , geometry
Let $T$ be a non-isosceles triangle and $n \ge 4$ an integer . Prove that you can divide $T$ in $n$ triangles and draw in each of them an inner bisector so that those $n$ bisectors are parallel.

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.

2004 Kazakhstan National Olympiad, 2
Tags: combinatorial geometry , combinatorics
A [i]zigzag [/i] is a polyline on a plane formed from two parallel rays and a segment connecting the origins of these rays. What is the maximum number of parts a plane can be split into using $ n $ zigzags?

2020 Ukrainian Geometry Olympiad - December, 2
Tags: combinatorics , isosceles , combinatorial geometry , geometry
On a straight line lie $100$ points and another point outside the line. Which is the biggest the number of isosceles triangles can be formed from the vertices of these $101$ points?