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: 121

2014 Junior Balkan Team Selection Tests - Romania, 3
Tags: point , combinatorics , distance , combinatorial geometry
Consider six points in the interior of a square of side length $3$. Prove that among the six points, there are two whose distance is less than $2$.

2009 All-Russian Olympiad Regional Round, 11.5
Tags: point , combinatorial geometry , line , combinatorics
We drew several straight lines on the plane and marked all of them intersection points. How many lines could be drawn? if one point is marked on one of the drawn lines, on the other - three, and on the third - five? Find all possible options and prove that there are no others.

1990 All Soviet Union Mathematical Olympiad, 513
Tags: point , edge , graph , combinatorial geometry , combinatorics
A graph has $30$ points and each point has $6$ edges. Find the total number of triples such that each pair of points is joined or each pair of points is not joined.

2020 Federal Competition For Advanced Students, P2, 2
Tags: coloring , point , combinatorics
In the plane there are $2020$ points, some of which are black and the rest are green. For every black point, the following applies: [i]There are exactly two green points that represent the distance $2020$ from that black point. [/i] Find the smallest possible number of green dots. (Walther Janous)

1953 Moscow Mathematical Olympiad, 251
Tags: point , combinatorial geometry , convex polygon , circles
On a circle, distinct points $A_1, ... , A_{16}$ are chosen. Consider all possible convex polygons all of whose vertices are among $A_1, ... , A_{16}$ . These polygons are divided into $2$ groups, the first group comprising all polygons with $A_1$ as a vertex, the second group comprising the remaining polygons. Which group is more numerous?

2012 Tournament of Towns, 2
Tags: point , circles , combinatorial geometry , combinatorics
One hundred points are marked inside a circle, with no three in a line. Prove that it is possible to connect the points in pairs such that all fifty lines intersect one another inside the circle.

2017 Saudi Arabia Pre-TST + Training Tests, 8
Tags: point , combinatorics , combinatorial geometry
There are $2017$ points on the plane, no three of them are collinear. Some pairs of the points are connected by $n$ segments. Find the smallest value of $n$ so that there always exists two disjoint segments in any case.

2000 Switzerland Team Selection Test, 15
Tags: combinatorics , geometric inequalities , point , geometry , combinatorial geometry
Let $S = \{P_1,P_2,...,P_{2000}\}$ be a set of $2000$ points in the interior of a circle of radius $1$, one of which at its center. For $i = 1,2,...,2000$ denote by $x_i$ the distance from $P_i$ to the closest point $P_j \ne P_i$. Prove that $x_1^2 +x_2^2 +...+x_{2000}^2<9$ .

2000 BAMO, 4
Tags: combinatorial geometry , point , combinatorics , distance
Prove that there exists a set $S$ of $3^{1000}$ points in the plane such that for each point $P$ in $S$, there are at least $2000$ points in $S$ whose distance to $P$ is exactly $1$ inch.

1998 Belarusian National Olympiad, 7
Tags: point , combinatorics , combinatorial geometry
On the plane $n+1$ points are marked, no three of which lie on one straight line. For what natural $k$ can they be connected by segments so that for any $n$ marked points there are exactly $k$ segments with ends at these points?

Kyiv City MO Juniors Round2 2010+ geometry, 2019.7.31
Tags: point , construction , coordinate geometry , geometry , analytic geometry
The teacher drew a coordinate plane on the board and marked some points on this plane. Unfortunately, Vasya's second-grader, who was on duty, erased almost the entire drawing, except for two points $A (1, 2)$ and $B (3,1)$. Will the excellent Andriyko be able to follow these two points to construct the beginning of the coordinate system point $O (0, 0)$? Point A on the board located above and to the left of point $B$.

1984 Tournament Of Towns, (072) 3
Tags: point , segment , combinatorial geometry , geometry
On a plane there is a finite set of $M$ points, no three of which are collinear . Some points are joined to others by line segments, with each point connected to no more than one line segment . If we have a pair of intersecting line segments $AB$ and $CD$ we decide to replace them with $AC$ and $BD$, which are opposite sides of quadrilateral $ABCD$. In the resulting system of segments we decide to perform a similar substitution, if possible, and so on . Is it possible that such substitutions can be carried out indefinitely? (V.E. Kolosov)

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 fi fty segments intersect one another?

VI Soros Olympiad 1999 - 2000 (Russia), 10.2
Tags: combinatorial geometry , point , geometry , combinatorics
$37$ points are arbitrarily marked on the plane. Prove that among them there must be either two points at a distance greater than $6$, or two points at a distance less than $1.5$.

1979 Chisinau City MO, 181
Tags: combinatorial geometry , point , collinear , geometry
Prove that if every line connecting any two points of some finite set of points of the plane contains at least one more point of this set, then all points of the set lie on one straight line.

2004 Chile National Olympiad, 2
Tags: combinatorial geometry , point , coloring , combinatorics
Every point on a line is painted either red or blue. Prove that there always exist three points $A,B,C$ that are painted the same color and are such that the point $B$ is the midpoint of the segment $AC$.

1982 All Soviet Union Mathematical Olympiad, 333
Tags: point , diameter , combinatorics , combinatorial geometry
$3k$ points are marked on the circumference. They divide it onto $3k$ arcs. Some $k$ of them have length $1$, other $k$ of them have length $2$, the rest $k$ of them have length $3$. Prove that some two of the marked points are the ends of one diameter.

2009 Mathcenter Contest, 3
Tags: combinatorial geometry , combinatorics , point
Prove that for each $k$ points in the plane, no three collinear and having integral distances from each other. If we have an infinite set of points with integral distances from each other, then all points are collinear. [i](Anonymous314)[/i] PS. wording needs to be fixed , [url=http://www.mathcenter.net/forum/showthread.php?t=7288]source[/url]

1986 China Team Selection Test, 4
Tags: point , chord , combinatorics
Mark $4 \cdot k$ points in a circle and number them arbitrarily with numbers from $1$ to $4 \cdot k$. The chords cannot share common endpoints, also, the endpoints of these chords should be among the $4 \cdot k$ points. [b]i.[/b] Prove that $2 \cdot k$ pairwisely non-intersecting chords can be drawn for each of whom its endpoints differ in at most $3 \cdot k - 1$. [b]ii.[/b] Prove that the $3 \cdot k - 1$ cannot be improved.

2001 Junior Balkan Team Selection Tests - Moldova, 1
Tags: combinatorial geometry , coloring , point , combinatorics
On a circle we consider a set $M$ consisting of $n$ ($n \ge 3$) points, of which only one is colored red. Determine of which polygons inscribed in a circle having the vertices in the set $M$ are more: those that contain the red dot or those that do not contain those points? How many more are there than others?

2016 Kazakhstan National Olympiad, 5
Tags: combinatorial geometry , coloring , point , combinatorics
$101$ blue and $101$ red points are selected on the plane, and no three lie on one straight line. The sum of the pairwise distances between the red points is $1$ (that is, the sum of the lengths of the segments with ends at red points), the sum of the pairwise distances between the blue ones is also $1$, and the sum of the lengths of the segments with the ends of different colors is $400$. Prove that you can draw a straight line separating everything red dots from all blue ones.

2013 Junior Balkan Team Selection Tests - Romania, 2
Tags: point , lattice , coloring , combinatorics
Let $M$ be the set of integer coordinate points situated on the line $d$ of real numbers. We color the elements of M in black or white. Show that at least one of the following statements is true: (a) there exists a finite subset $F \subset M$ and a point $M \in d$ so that the elements of the set $M - F$ that are lying on one of the rays determined by $M$ on $d$ are all white, and the elements of $M - F$ that are situated on the opposite ray are all black, (b) there exists an infinite subset $S \subset M$ and a point $T \in d$ so that for each $A \in S$ the reflection of A about $T$ belongs to $S$ and has the same color as $A$

1998 Estonia National Olympiad, 5
Tags: combinatorial geometry , point , coloring , circles , combinatorics
A circle is divided into $n$ equal arcs by $n$ points. Assume that, no matter how we color the $n$ points in two colors, there always exists an axis of symmetry of the set of points such that any two of the $n$ points which are symmetric with respect to that axis have the same color. Find all possible values of $n$.

1969 Polish MO Finals, 6
Tags: point , combinatorial geometry , geometry
Given a set $n$ of points in the plane that are not contained in a single straight line. Prove that there exists a circle passing through at least three of these points, inside which there are none of the remaining points of the set.

2022 Sharygin Geometry Olympiad, 8.7
Tags: point , combinatorial geometry , geometry , combinatorics
Ten points on a plane a such that any four of them lie on the boundary of some square. Is obligatory true that all ten points lie on the boundary of some square?