Olympiad problems

1980 Austrian-Polish Competition, 8

Let $S$ be a set of 1980 points in the plane such that the distance between every pair of them is at least 1. Prove that $S$ has a subset of 220 points such that the distance between every pair of them is at least $\sqrt{3}.$

2024 Junior Balkan Team Selection Tests - Romania, P4

Tags: combinatorial geometry , combinatorics

Let $n\geqslant 2$ be an integer and $A{}$ a set of $n$ points in the plane. Find all integers $1\leqslant k\leqslant n-1$ with the following property: any two circles $C_1$ and $C_2$ in the plane such that $A\cap\text{Int}(C_1)\neq A\cap\text{Int}(C_2)$ and $|A\cap\text{Int}(C_1)|=|A\cap\text{Int}(C_2)|=k$ have at least one common point. [i]Cristi Săvescu[/i]

KoMaL A Problems 2024/2025, A. 887

Tags: grid , combinatorial geometry , geometry

A non self-intersecting polygon is given in a Cartesian coordinate system such that its perimeter contains no lattice points, and its vertices have no integer coordinates. A point is called semi-integer if exactly one of its coordinates is an integer. Let $P_1, P_2,\ldots, P_k$ denote the semi-integer points on the perimeter of the polygon. Let ni denote the floor of the non-integer coordinate of $P_i$. Prove that integers $n_1,n_2,\ldots ,n_k$ can be divided into two groups with the same sum. [i]Proposed by Áron Bán-Szabó, Budapest[/i]

1989 All Soviet Union Mathematical Olympiad, 506

Tags: combinatorics , geometry , combinatorial geometry

Two walkers are at the same altitude in a range of mountains. The path joining them is piecewise linear with all its vertices above the two walkers. Can they each walk along the path until they have changed places, so that at all times their altitudes are equal?

1999 All-Russian Olympiad, 6

Tags: combinatorial geometry , geometry , rotation

Three convex polygons are given on a plane. Prove that there is no line cutting all the polygons if and only if each of the polygons can be separated from the other two by a line.

2017 Saudi Arabia Pre-TST + Training Tests, 6

Tags: combinatorial geometry , convex , combinatorics

A convex polygon is divided into some triangles. Let $V$ and $E$ be respectively the set of vertices and the set of egdes of all triangles (each vertex in $V$ may be some vertex of the polygon or some point inside the polygon). The polygon is said to be [i]good [/i] if the following conditions hold: i. There are no $3$ vertices in $V$ which are collinear. ii. Each vertex in $V$ belongs to an even number of edges in $E$. Find all good polygon.

2020/2021 Tournament of Towns, P5

Tags: combinatorial geometry , combinatorics

Does there exist a rectangle which can be cut into a hundred rectangles such that all of them are similar to the original one but no two are congruent? [i]Mikhail Murashkin[/i]

2007 Finnish National High School Mathematics Competition, 3

Tags: combinatorial geometry , geometry

There are five points in the plane, no three of which are collinear. Show that some four of these points are the vertices of a convex quadrilateral.

2024 Middle European Mathematical Olympiad, 2

Tags: question , combinatorics , guessing game , polygon , combinatorial geometry

There is a rectangular sheet of paper on an infinite blackboard. Marvin secretly chooses a convex $2024$-gon $P$ that lies fully on the piece of paper. Tigerin wants to find the vertices of $P$. In each step, Tigerin can draw a line $g$ on the blackboard that is fully outside the piece of paper, then Marvin replies with the line $h$ parallel to $g$ that is the closest to $g$ which passes through at least one vertex of $P$. Prove that there exists a positive integer $n$, independent of the choice of the polygon, such that Tigerin can always determine the vertices of $P$ in at most $n$ steps.

1999 Argentina National Olympiad, 5

Tags: combinatorics , rectangle , geometry , combinatorial geometry

A rectangle-shaped puzzle is assembled with $2000$ pieces that are all equal rectangles, and similar to the large rectangle, so that the sides of the small rectangles are parallel to those of the large one. The shortest side of each piece measures $1$. Determine what is the minimum possible value of the area of the large rectangle.

1987 Flanders Math Olympiad, 1

Tags: combinatorics , combinatorial geometry

A rectangle $ABCD$ is given. On the side $AB$, $n$ different points are chosen strictly between $A$ and $B$. Similarly, $m$ different points are chosen on the side $AD$. Lines are drawn from the points parallel to the sides. How many rectangles are formed in this way? (One possibility is shown in the figure.) [img]https://cdn.artofproblemsolving.com/attachments/0/1/dcf48e4ce318fdcb8c7088a34fac226e26e246.png[/img]

2021 China Team Selection Test, 6

Tags: combinatorial geometry , combinatorics , geometry

Proof that there exist constant $\lambda$, so that for any positive integer $m(\ge 2)$, and any lattice triangle $T$ in the Cartesian coordinate plane, if $T$ contains exactly one $m$-lattice point in its interior(not containing boundary), then $T$ has area $\le \lambda m^3$. PS. lattice triangles are triangles whose vertex are lattice points; $m$-lattice points are lattice points whose both coordinates are divisible by $m$.

1990 Poland - Second Round, 6

Tags: combinatorics , convex polygon , area , combinatorial geometry , geometry

For any convex polygon $ W $ with area 1, let us denote by $ f(W) $ the area of the convex polygon whose vertices are the centers of all sides of the polygon $ W $. For each natural number $ n \geq 3 $, determine the lower limit and the upper limit of the set of numbers $ f(W) $ when $ W $ runs through the set of all $ n $ convex angles with area 1.

1971 Poland - Second Round, 4

Tags: combinatorics , combinatorial geometry , geometry

On the plane there is a finite set of points $Z$ with the property that no two distances of the points of the set $Z$ are equal. We connect the points $ A, B $ belonging to $ Z $ if and only if $ A $ is the point closest to $ B $ or $ B $ is the point closest to $ A $. Prove that no point in the set $Z$ will be connected to more than five others.

2012 Switzerland - Final Round, 8

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

Consider a cube and two of its vertices $A$ and $B$, which are the endpoints of a face diagonal. A [i]path [/i] is a sequence of cube angles, each step of one angle along a cube edge is walked to one of the three adjacent angles. Let $a$ be the number of paths of length $2012$ that starts at point $A$ and ends at $A$ and let b be the number of ways of length $2012$ that starts in $A$ and ends in $B$. Decide which of the two numbers $a$ and $b$ is the larger.

2012 Danube Mathematical Competition, 1

Tags: lattice points , covering , square , combinatorics , combinatorial geometry

Given a positive integer $n$, determine the maximum number of lattice points in the plane a square of side length $n +\frac{1}{2n+1}$ may cover.

1989 Tournament Of Towns, (221) 5

Tags: line , combinatorics , combinatorial geometry

We are given $N$ lines ($N > 1$ ) in a plane, no two of which are parallel and no three of which have a point in common. Prove that it is possible to assign, to each region of the plane determined by these lines, a non-zero integer of absolute value not exceeding $N$ , such that the sum of the integers o n either side of any of the given lines is equal to $0$ . (S . Fomin, Leningrad)

1969 Bulgaria National Olympiad, Problem 3

Tags: geometry , combinatorial geometry , combinatorics

Some of the points in the plane are white and some are blue (every point of the plane is either white or blue). Prove that for every positive number $r$: (a) there are at least two points with different color such that the distance between them is equal to $r$; (b) there are at least two points with the same color and the distance between them is equal to $r$; (c) will the statements above be true if the plane is replaced with the real line?

1992 Tournament Of Towns, (353) 2

Tags: coloring , combinatorics , combinatorial geometry

For which values of $n$ is it possible to construct an $n$ by $n$ by $n$ cube with $n^3$ unit cubes, each of which is black or white, such that each cube shares a common face with exactly three cubes of the opposite colour? (S Tokarev)

2020 Simon Marais Mathematics Competition, A1

Tags: combinatorics , combinatorial geometry

There are $1001$ points in the plane such that no three are collinear. The points are joined by $1001$ line segments such that each point is an endpoint of exactly two of the line segments. Prove that there does not exist a straight line in the plane that intersects each of the $1001$ segments in an interior point. [i]An interior point of a line segment is a point of the line segment that is not one of the two endpoints.[/i]

2022 Swedish Mathematical Competition, 6

Tags: combinatorics , combinatorial geometry

Bengt wants to put out crosses and rings in the squares of an $n \times n$-square, so that it is exactly one ring and exactly one cross in each row and in each column, and no more than one symbol in each box. Mona wants to stop him by setting a number in advance ban on crosses and a number of bans on rings, maximum one ban in each square. She want to use as few bans as possible of each variety. To succeed in preventing Bengt, how many prohibitions she needs to use the least of the kind of prohibitions she uses the most of?

2016 Putnam, B3

Tags: putnam geometry , combinatorial geometry

Suppose that $S$ is a finite set of points in the plane such that the area of triangle $\triangle ABC$ is at most $1$ whenever $A,B,$ and $C$ are in $S.$ Show that there exists a triangle of area $4$ that (together with its interior) covers the set $S.$

2018 Estonia Team Selection Test, 2

Tags: combinatorial geometry , square table , combinatorics , grid

Find the greatest number of depicted pieces composed of $4$ unit squares that can be placed without overlapping on an $n \times n$ grid (where n is a positive integer) in such a way that it is possible to move from some corner to the opposite corner via uncovered squares (moving between squares requires a common edge). The shapes can be rotated and reflected. [img]https://cdn.artofproblemsolving.com/attachments/b/d/f2978a24fdd737edfafa5927a8d2129eb586ee.png[/img]

2000 Swedish Mathematical Competition, 4

Tags: lattice , area , geometric inequalities , 3d geometry , geometry , combinatorial geometry

The vertices of a triangle are three-dimensional lattice points. Show that its area is at least $\frac12$.

1948 Moscow Mathematical Olympiad, 142

Tags: distance , combinatorial geometry , combinatorics

Find all possible arrangements of $4$ points on a plane, so that the distance between each pair of points is equal to either $a$ or $b$. For what ratios of $a : b$ are such arrangements possible?