Olympiad problems

1996 North Macedonia National Olympiad, 4

A polygon is called [i]good [/i] if it satisfies the following conditions: (i) All its angles are in $(0,\pi)$ or in $(\pi ,2\pi)$, (ii) It is not self-intersecing, (iii) For any three sides, two are parallel and equal. Find all $n$ for which there exists a [i]good [/i] $n$-gon.

2024 Romanian Master of Mathematics, 3

Tags: geometry , combinatorics , linear algebra , combinatorial geometry , set

Given a positive integer $n$, a collection $\mathcal{S}$ of $n-2$ unordered triples of integers in $\{1,2,\ldots,n\}$ is [i]$n$-admissible[/i] if for each $1 \leq k \leq n - 2$ and each choice of $k$ distinct $A_1, A_2, \ldots, A_k \in \mathcal{S}$ we have $$ \left|A_1 \cup A_2 \cup \cdots A_k \right| \geq k+2.$$ Is it true that for all $n > 3$ and for each $n$-admissible collection $\mathcal{S}$, there exist pairwise distinct points $P_1, \ldots , P_n$ in the plane such that the angles of the triangle $P_iP_jP_k$ are all less than $61^{\circ}$ for any triple $\{i, j, k\}$ in $\mathcal{S}$? [i]Ivan Frolov, Russia[/i]

1990 Tournament Of Towns, (276) 4

Tags: combinatorics , combinatorial geometry , geometry

We have “bricks” made in the following way: we take a unit cube and glue to three of its faces which have a common vertex three more cubes in such a way that the faces glued together coincide. Is it possible to construct from these bricks an $11 \times 12 \times 13$ box? (A Andjans, Riga )

2012 Tournament of Towns, 2

Tags: point , combinatorial geometry , combinatorics

One hundred points are marked in the plane, with no three in a line. Is it always possible to connect the points in pairs such that all fifty segments intersect one another?

May Olympiad L2 - geometry, 2005.1

Tags: combinatorics , combinatorial geometry

The enemy ship has landed on a $9\times 9$ board that covers exactly $5$ squares of the board, like this: [img]https://cdn.artofproblemsolving.com/attachments/2/4/ae5aa95f5bb5e113fd5e25931a2bf8eb872dbe.png[/img] The ship is invisible. Each defensive missile covers exactly one square, and destroys the ship if it hits one of the $5$ squares that it occupies. Determine the minimum number of missiles needed to destroy the enemy ship with certainty .

1996 Chile National Olympiad, 3

Tags: combinatorial geometry , combinatorics , line

Let $n> 2$ be a natural. Given $2n$ points in the plane, no $3$ are collinear. What is the maximum number of lines that can be drawn between them, without forming a triangle? [hide=original wording]Sea n > 2 un natural. Dados 2n puntos en el plano, tres a tres no colineales, Cual es el numero maximo de trazos que pueden dibujarse entre ellos, sin formar un triangulo?[/hide]

2018 Irish Math Olympiad, 4

Tags: rectangle , combinatorics , combinatorial geometry

We say that a rectangle with side lengths $a$ and $b$ [i]fits inside[/i] a rectangle with side lengths $c$ and $d$ if either ($a \le c$ and $b \le d$) or ($a \le d$ and $b \le c$). For instance, a rectangle with side lengths $1$ and $5$ [i]fits inside[/i] another rectangle with side lengths $1$ and $5$, and also [i]fits inside[/i] a rectangle with side lengths $6$ and $2$. Suppose $S$ is a set of $2019$ rectangles, all with integer side lengths between $1$ and $2018$ inclusive. Show that there are three rectangles $A$, $B$, and $C$ in $S$ such that $A$ fits inside $B$, and $B$ [i]fits inside [/i]$C$.

2013 Tournament of Towns, 3

Tags: geometry , lattice points , parallel , combinatorial geometry

A point in the plane is called a node if both its coordinates are integers. Consider a triangle with vertices at nodes containing at least two nodes inside. Prove that there exists a pair of internal nodes such that a straight line connecting them either passes through a vertex or is parallel to side of the triangle.

1992 Romania Team Selection Test, 8

Tags: combinatorial geometry , combinatorics

Let $m,n \ge 2$ be integers. The sides $A_{00}A_{0m}$ and $A_{nm}A_{n0}$ of a convex quadrilateral $A_{00}A_{0m}A_{nm}A_{n0}$ are divided into $m$ equal segments by points $A_{0j}$ and $A_{nj}$ respectively ($j = 1,...,m-1$). The other two sides are divided into $n$ equal segments by points $A_{i0}$ and $A_{im}$ ($i = 1,...,n -1$). Denote by $A_{ij}$ the intersection of lines $A_{0j}A{nj}$ and $A_{i0}A_{im}$, by $S_{ij}$ the area of quadrilateral $A_{ij}A_{i, j+1}A_{i+1, j+1}A_{i+1, j}$ and by $S$ the area of the big quadrilateral. Show that $S_{ij} +S_{n-1-i,m-1-j} = \frac{2S}{mn}$

2020 Colombia National Olympiad, 2

Tags: combinatorics , combinatorial geometry , coloring

Given a regular $n$-sided polygon with $n \ge 3$. Maria draws some of its diagonals in such a way that each diagonal intersects at most one of the other diagonals drawn in the interior of the polygon. Determine the maximum number of diagonals that Maria can draw in such a way. Note: Two diagonals can share a vertex of the polygon. Vertices are not part of the interior of the polygon.

2011 NZMOC Camp Selection Problems, 6

Tags: combinatorics , combinatorial geometry

Consider the set $G$ of $2011^2$ points $(x, y)$ in the plane where $x$ and $y$ are both integers between $ 1$ and $2011$ inclusive. Let $A$ be any subset of $G$ containing at least $4\times 2011\times \sqrt{2011}$ points. Show that there are at least $2011^2$ parallelograms whose vertices lie in $A$ and all of whose diagonals meet at a single point.

1974 Dutch Mathematical Olympiad, 1

Tags: combinatorial geometry , geometric inequality , area , geometry

A convex quadrilateral with area $1$ is divided into four quadrilaterals divided by connecting the midpoints of the opposite sides. Prove that each of those four quadrilaterals has area $< \frac38$.

1974 Chisinau City MO, 78

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

Each point of the sphere of radius $r\ge 1$ is colored in one of $n$ colors ($n \ge 2$), and for each color there is a point on the sphere colored in this color. Prove that there are points $A_i$, $B_i$, $i= 1, ..., n$ on the sphere such that the colors of the points $A_1, ..., A_n$ are pairwise different and the color of the point $B_i$ at a distance of $1$ from $A_i$ is different from the color of the point $A_1, i= 1, ..., n$

2019 Baltic Way, 15

Tags: geometry , combinatorial geometry

Let $n \geq 4$, and consider a (not necessarily convex) polygon $P_1P_2\hdots P_n$ in the plane. Suppose that, for each $P_k$, there is a unique vertex $Q_k\ne P_k$ among $P_1,\hdots, P_n$ that lies closest to it. The polygon is then said to be [i]hostile[/i] if $Q_k\ne P_{k\pm 1}$ for all $k$ (where $P_0 = P_n$, $P_{n+1} = P_1$). (a) Prove that no hostile polygon is convex. (b) Find all $n \geq 4$ for which there exists a hostile $n$-gon.

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]

1992 Miklós Schweitzer, 9

Tags: combinatorial geometry , euclidean geometry , convex geometry

Let K be a bounded, d-dimensional convex polyhedron that is not simplex and P is a point on K. Show that if vertices $P_1 , ..., P_k$ are not all on the same face of K, then one of them can be omitted so that the convex hull of the remaining vertices of K still contains P. [hide=note]caratheodory's theorem might be useful. [/hide]

1969 IMO Longlists, 68

Tags: geometry , point set , convex quadrilateral , combinatorial geometry

$(USS 5)$ Given $5$ points in the plane, no three of which are collinear, prove that we can choose $4$ points among them that form a convex quadrilateral.

2012 QEDMO 11th, 8

Tags: point , combinatorial geometry , combinatorics

Prove that there are $2012$ points in the plane, none of which are three on one straight line and in pairs have integer distances .

1978 Germany Team Selection Test, 6

Tags: combinatorics , lattice points , combinatorial geometry , circles

A lattice point in the plane is a point both of whose coordinates are integers. Each lattice point has four neighboring points: upper, lower, left, and right. Let $k$ be a circle with radius $r \geq 2$, that does not pass through any lattice point. An interior boundary point is a lattice point lying inside the circle $k$ that has a neighboring point lying outside $k$. Similarly, an exterior boundary point is a lattice point lying outside the circle $k$ that has a neighboring point lying inside $k$. Prove that there are four more exterior boundary points than interior boundary points.

1982 Kurschak Competition, 1

Tags: geometry , 3d geometry , combinatorial geometry , lattice

A cube of integral dimensions is given in space so that all four vertices of one of the faces are lattice points. Prove that the other four vertices are also lattice points.

1998 Tournament Of Towns, 4

Tags: diagonal , combinatorial geometry , concurrency , combinatorics

All the diagonals of a regular $25$-gon are drawn. Prove that no $9$ of the diagonals pass through one interior point of the $25$-gon. (A Shapovalov)

2015 QEDMO 14th, 1

Tags: combinatorics , combinatorial geometry

Let $n$ be a natural number. A regular hexagon with edge length $n$ gets split into equilateral exploded triangles whose edges are $1$ in length and parallel to one side of the hexagon. Find the number of regular hexagons, the angles of which are all angles of these triangles are.

2019 Swedish Mathematical Competition, 4

Tags: combinatorial geometry , combinatorics , circles

Let $\Omega$ be a circle disk with radius $1$. Determine the minimum $r$ that has the following property: You can select three points on $\Omega$ so that each circle disk located in $\Omega$ and has a radius greater than $r$ contains at least one of the three points.

2013 Swedish Mathematical Competition, 4

Tags: combinatorics , combinatorial geometry , geometry

A robotic lawnmower is located in the middle of a large lawn. Due a manufacturing defect, the robot can only move straight ahead and turn in directions that are multiples of $60^o$. A fence must be set up so that it delimits the entire part of the lawn that the robot can get to, by traveling along a curve with length no more than $10$ meters from its starting position, given that it is facing north when it starts. How long must the fence be?

2008 Switzerland - Final Round, 4

Tags: combinatorial geometry , combinatorics

Consider three sides of an $n \times n \times n$ cube that meet at one of the corners of the cube. For which $n$ is it possible to use this completely and without overlapping to cover strips of paper of size $3 \times 1$? The paper strips can also do this glued over the edges between these cube faces.