Found problems: 115
1998 All-Russian Olympiad Regional Round, 10.3
Prove that from any finite set of points on the plane, you can remove a point from the bottom in such a way that the remaining set can be split into two parts of smaller diameter. (Diameter is the maximum distance between points of the set.)
[hide=original wording]Докажите, что из любого конечного множества точек на плоскости можно так удалитьо дну точку, что оставшееся множество можно разбить на две части меньшего диаметра. (Диаметр—это максимальное расстояние между точками множества.)[/hide]
2014 Chile National Olympiad, 3
In the plane there are $2014$ plotted points, such that no $3$ are collinear. For each pair of plotted points, draw the line that passes through them. prove that for every three of marked points there are always two that are separated by an amount odd number of lines.
2009 Abels Math Contest (Norwegian MO) Final, 3b
Show for any positive integer $n$ that there exists a circle in the plane such that there are exactly $n$ grid points within the circle. (A grid point is a point having integer coordinates.)
2009 Mathcenter Contest, 3
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]
2010 Junior Balkan Team Selection Tests - Romania, 5
Let $n$ be a non-zero natural number, $n \ge 5$. Consider $n$ distinct points in the plane, each colored or white, or black. For each natural $k$ , a move of type $k, 1 \le k <\frac {n} {2}$, means selecting exactly $k$ points and changing their color. Determine the values of $n$ for which, whatever $k$ and regardless of the initial coloring, there is a finite sequence of $k$ type moves, at the end of which all points have the same color.
2008 Chile National Olympiad, 4
Three colors are available to paint the plane. If each point in the plane is assigned one of these three colors, prove that there is a segment of length $1$ whose endpoints have the same color.
2004 Chile National Olympiad, 2
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 $AC$.
1977 Czech and Slovak Olympiad III A, 5
Let $A_1,\ldots,A_n$ be different collinear points. Every point is dyed by one of four colors and every of these colors is used at least once. Show that there is a line segment where two colors are used exactly once and the other two are used at least once.
2018-IMOC, G1
Given an integer $n \ge 3$. Find the largest positive integer $k $ with the following property:
For $n$ points in general position, there exists $k$ ways to draw a non-intersecting polygon with those $n$ points as it’s vertices.
[hide=Different wording]Given $n$, find the maximum $k$ so that for every general position of $n$ points , there are at least $k$ ways of connecting the points to form a polygon.[/hide]
1953 Moscow Mathematical Olympiad, 251
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?
2009 Chile National Olympiad, 6
There are $n \ge 6$ green points in the plane, such that no $3$ of them are collinear. Suppose further that $6$ of these points are the vertices of a convex hexagon. Prove that there are $5$ green points that form a pentagon that does not contain any other green point inside.
2016 Israel National Olympiad, 4
In the beginning, there is a circle with three points on it. The points are colored (clockwise): Green, blue, red. Jonathan may perform the following actions, as many times as he wants, in any order:
[list]
[*] Choose two adjacent points with [u]different[/u] colors, and add a point between them with one of the two colors only.
[*] Choose two adjacent points with [u]the same[/u] color, and add a point between them with any of the three colors.
[*] Choose three adjacent points, at least two of them having the same color, and delete the middle point.
[/list]
Can Jonathan reach a state where only three points remain on the circle, colored (clockwise): Blue, green, red?
2016 Kazakhstan National Olympiad, 5
$101$ 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.
Estonia Open Junior - geometry, 2016.2.5
On the plane three different points $P, Q$, and $R$ are chosen. It is known that however one chooses another point $X$ on the plane, the point $P$ is always either closer to $X$ than the point $Q$ or closer to $X$ than the point $R$. Prove that the point $P$ lies on the line segment $QR$.
2012 Tournament of Towns, 2
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.
1982 All Soviet Union Mathematical Olympiad, 333
$3k$ 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.
2007 Peru MO (ONEM), 2
Assuming that each point of a straight line is painted red or blue, arbitrarily, show that it is always possible to choose three points $A, B$ and $C$ in such a way straight, that are painted the same color and that: $$\frac{AB}{1}=\frac{BC}{2}=\frac{AC}{3}.$$
2016 Singapore Junior Math Olympiad, 5
Determine the minimum number of lines that can be drawn on the plane so that they intersect in exactly $200$ distinct points.
(Note that for $3$ distinct points, the minimum number of lines is $3$ and for $4$ distinct points, the minimum is $4$)
1984 Tournament Of Towns, (072) 3
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)
2000 BAMO, 4
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.
1981 Brazil National Olympiad, 4
A graph has $100$ points. Given any four points, there is one joined to the other three. Show that one point must be joined to all $99$ other points. What is the smallest number possible of such points (that are joined to all the others)?
1998 German National Olympiad, 1
Find all possible numbers of lines in a plane which intersect in exactly $37$ points.
VI Soros Olympiad 1999 - 2000 (Russia), 10.2
$37$ 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 $1.5$.
1994 Tuymaada Olympiad, 1
World Cup in America introduced a new point system. For a victory $3$ points are given, for a draw $1$ point and for defeat $0$ points. In the preliminary games, the teams are divided into groups of $4$ teams. In groups, teams play with each other, once, then in accordance with the points scored $a,b,c$ and $d$ ($a>b>c>d$) teams take the first, second, third and fourth place in their groups. Give all possible options for the distribution points $a,b,c$ and $d$
1982 Tournament Of Towns, (026) 4
(a) $10$ points dividing a circle into $10$ equal arcs are connected in pairs by $5$ chords.
Is it necessary that two of these chords are of equal length?
(b) $20$ points dividing a circle into $20$ equal arcs are connected in pairs by $10$ chords.
Prove that among these $10$ chords there are two chords of equal length.
(VV Proizvolov, Moscow)