Olympiad problems

Kvant 2022, M2708 b)

Do there exist 100 points on the plane such that the pairwise distances between them are pairwise distinct consecutive integer numbers larger than 2022?

2018 Irish Math Olympiad, 4

Tags: combinatorics , combinatorial geometry , rectangle

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]$C$.

2012 All-Russian Olympiad, 2

Tags: combinatorics , combinatorial geometry

A regular $2012$-gon is inscribed in a circle. Find the maximal $k$ such that we can choose $k$ vertices from given $2012$ and construct a convex $k$-gon without parallel sides.

1961 Kurschak Competition, 1

Tags: combinatorial geometry , combinatorics , point , ratio

Given any four distinct points in the plane, show that the ratio of the largest to the smallest distance between two of them is at least $\sqrt2$.

2009 IMAR Test, 2

Tags: geometry , combinatorial geometry , greatest common divisor , combinatorics , number theory

Of the vertices of a cube, $7$ of them have assigned the value $0$, and the eighth the value $1$. A [i]move[/i] is selecting an edge and increasing the numbers at its ends by an integer value $k > 0$. Prove that after any finite number of [i]moves[/i], the g.c.d. of the $8$ numbers at vertices is equal to $1$. Russian M.O.

2020 Swedish Mathematical Competition, 6

Tags: combinatorial geometry , combinatorics

A finite set of [i]axis parallel [/i]cubes in space has the property of each point of the room is located in a maximum of M different cubes. Show that you can divide the amount of cubes in $8 (M - 1) + 1$ subsets (or less) with the property that the cubes in each subset lacks common points. (An axis parallel cube is a cube whose edges are parallel to the coordinate axes.)

1991 All Soviet Union Mathematical Olympiad, 537

Tags: combinatorial geometry , line , integer sides

Four lines in the plane intersect in six points. Each line is thus divided into two segments and two rays. Is it possible for the eight segments to have lengths $1, 2, 3, ... , 8$? Can the lengths of the eight segments be eight distinct integers?

1962 Leningrad Math Olympiad, 7.5*

Tags: combinatorial geometry , geometry , combinatorics

The circle is divided into $49$ areas so that no three areas touch at one point. The resulting “map” is colored in three colors so that no two adjacent areas have the same color. The border of two areas is considered to be colored in both colors. Prove that on the circle there are two diametrically opposite points, colored in one color.

2020 Estonia Team Selection Test, 2

Tags: geometry , combinatorial geometry , combinatorics , angle , min

Let $n$ be an integer, $n \ge 3$. Select $n$ points on the plane, none of which are three on the same line. Consider all triangles with vertices at selected points, denote the smallest of all the interior angles of these triangles by the variable $\alpha$. Find the largest possible value of $\alpha$ and identify all the selected $n$ point placements for which the max occurs.

2002 All-Russian Olympiad Regional Round, 10.2

Tags: combinatorics , combinatorial geometry , lattice points , convex polygon

A convex polygon on a plane contains at least $m^2+1$ points with integer coordinates. Prove that it contains $m+1$ points with integer coordinates that lie on the same line.

Kvant 2021, M2664

Tags: combinatorics , combinatorial geometry

The point $O{}$ is given in the plane. Find all natural numbers $n{}$ for which $n{}$ points in the plane can be colored red, so that for any two red points $A{}$ and $B{}$ there is a third red point $C{}$ is such that $O{}$ lies strictly inside the triangle $ABC$. [i]From the folklore[/i]

2001 Kazakhstan National Olympiad, 8

Tags: geometry , combinatorial geometry , distance

There are $ n \geq4 $ points on the plane, the distance between any two of which is an integer. Prove that there are at least $ \frac {1} {6} $ distances, each of which is divisible by $3$.

2020 Ukrainian Geometry Olympiad - April, 5

Tags: combinatorics , combinatorial geometry

The plane shows $2020$ straight lines in general position, that is, there are none three intersecting at one point but no two parallel. Let's say, that the drawn line $a$ [i]detaches [/i] the drawn line $b$ if all intersection points of line $b$ with the other drawn lines lie in one half plane wrt to line $a$ (given the most straightforward $a$). Prove that you can be guaranteed find two drawn lines $a$ and $b$ that $a$ detaches $b$, but $b$ does not detach $a$.

1957 Moscow Mathematical Olympiad, 358

Tags: geometry , combinatorial geometry , broken line , 3d geometry , perpendicular

The segments of a closed broken line in space are of equal length, and each three consecutive segments are mutually perpendicular. Prove that the number of segments is divisible by $6$.

2010 Korea Junior Math Olympiad, 8

Tags: combinatorics , combinatorial geometry , lattice paths , lattice points

In a rectangle with vertices $(0, 0), (0, 2), (n,0),(n, 2)$, ($n$ is a positive integer) find the number of longest paths starting from $(0, 0)$ and arriving at $(n, 2)$ which satisfy the following: $\bullet$ At each movement, you can move right, up, left, down by $1$. $\bullet$ You cannot visit a point you visited before. $\bullet$ You cannot move outside the rectangle.

1987 Polish MO Finals, 1

Tags: geometry , combinatorics , combinatorial geometry , square , distance , point

There are $n \ge 2$ points in a square side $1$. Show that one can label the points $P_1, P_2, ... , P_n$ such that $\sum_{i=1}^n |P_{i-1} - P_i|^2 \le 4$, where we use cyclic subscripts, so that $P_0$ means $P_n$.

2004 Germany Team Selection Test, 2

Tags: geometry , angle , combinatorial geometry , polygon , extremal combinatorics , right 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]

1981 Poland - Second Round, 5

Tags: geometry , combinatorics , combinatorial geometry , point

In the plane there are two disjoint sets $ A $ and $ B $, each of which consists of $ n $ points, and no three points of the set $ A \cup B $ lie on one straight line. Prove that there is a set of $ n $ disjoint closed segments, each of which has one end in the set $ A $ and the other in the set $ B $.

2007 Balkan MO Shortlist, C1

Tags: floor function , geometry , combinatorics , combinatorial geometry , geometric transformation , rotation

For a given positive integer $n >2$, let $C_{1},C_{2},C_{3}$ be the boundaries of three convex $n-$ gons in the plane , such that $C_{1}\cap C_{2}, C_{2}\cap C_{3},C_{1}\cap C_{3}$ are finite. Find the maximum number of points of the sets $C_{1}\cap C_{2}\cap C_{3}$.

2001 Chile National Olympiad, 1

Tags: geometry , combinatorics , perimeter , combinatorial geometry

$\bullet$ In how many ways can triangles be formed whose sides are integers greater than $50$ and less than $100$? $\bullet$ In how many of these triangles is the perimeter divisible by $3$?

2021 Polish Junior MO First Round, 6

Tags: combinatorics , combinatorial geometry , diagonal

In the convex $(2n+2) $-gon are drawn $n^2$ diagonals. Prove that one of these of diagonals cuts the $(2n+2)$ -gon into two polygons, each of which has an odd number vertices.

2015 JBMO Shortlist, C2

Tags: combinatorics , point , combinatorial geometry , distance

$2015$ points are given in a plane such that from any five points we can choose two points with distance less than $1$ unit. Prove that $504$ of the given points lie on a unit disc.

1991 China Team Selection Test, 3

Tags: induction , combinatorics , invariant , combinatorial geometry

$5$ points are given in the plane, any three non-collinear and any four non-concyclic. If three points determine a circle that has one of the remaining points inside it and the other one outside it, then the circle is said to be [i]good[/i]. Let the number of good circles be $n$; find all possible values of $n$.

1986 IMO Longlists, 7

Tags: geometry , point set , combinatorial geometry , straight lines

Let $f(n)$ be the least number of distinct points in the plane such that for each $k = 1, 2, \cdots, n$ there exists a straight line containing exactly $k$ of these points. Find an explicit expression for $f(n).$ [i]Simplified version.[/i] Show that $f(n)=\left[\frac{n+1}{2}\right]\left[\frac{n+2}{2}\right].$ Where $[x]$ denoting the greatest integer not exceeding $x.$

2021 Romanian Master of Mathematics, 5

Tags: combinatorics , geometry , combinatorial geometry

Let $n$ be a positive integer. The kingdom of Zoomtopia is a convex polygon with integer sides, perimeter $6n$, and $60^\circ$ rotational symmetry (that is, there is a point $O$ such that a $60^\circ$ rotation about $O$ maps the polygon to itself). In light of the pandemic, the government of Zoomtopia would like to relocate its $3n^2+3n+1$ citizens at $3n^2+3n+1$ points in the kingdom so that every two citizens have a distance of at least $1$ for proper social distancing. Prove that this is possible. (The kingdom is assumed to contain its boundary.) [i]Proposed by Ankan Bhattacharya, USA[/i]