Found problems: 121
1964 Poland - Second Round, 6
Prove that from any five points in the plane it is possible to choose three points that are not vertices of an acute triangle.
2002 Switzerland Team Selection Test, 1
In space are given $24$ points, no three of which are collinear. Suppose that there are exactly $2002$ planes determined by three of these points. Prove that there is a plane containing at least six points.
1949-56 Chisinau City MO, 58
On the plane $n$ points are chosen so that exactly $m$ of them lie on one straight line and no three points not included in these $m$ points lie on one straight line. What is the number of all lines, each of which contains at least two of these points?
2016 IMAR Test, 2
Given a positive integer $n$, does there exist a planar polygon and a point in its plane such that every line through that point meets the boundary of the polygon at exactly $2n$ points?
2008 Postal Coaching, 6
Suppose $n$ straight lines are in the plane so that there exist seven points such that any of these line passes through at least three of these points. Find the largest possible value of $n$.
1998 Rioplatense Mathematical Olympiad, Level 3, 6
Let $k$ be a fixed positive integer. For each $n = 1, 2,...,$ we will call [i]configuration [/i] of order $n$ any set of $kn$ points of the plane, which does not contain $3$ collinear, colored with $k$ given colors, so that there are $n$ points of each color. Determine all positive integers $n$ with the following property: in each configuration of order $n$, it is possible to select three points of each color, such that the $k$ triangles with vertices of the same color that are determined are disjoint in pairs.
1956 Moscow Mathematical Olympiad, 324
a) What is the least number of points that can be chosen on a circle of length $1956$, so that for each of these points there is exactly one chosen point at distance $1$, and exactly one chosen point at distance $2$ (distances are measured along the circle)?
b) On a circle of length $15$ there are selected $n$ points such that for each of them there is exactly one selected point at distance $1$ from it, and exactly one is selected point at distance $2$ from it. (All distances are measured along the circle.) Prove that $n$ is divisible by $10$.
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: $$\frac{AB}{1}=\frac{BC}{2}=\frac{AC}{3}.$$
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]
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)
2013 Junior Balkan Team Selection Tests - Romania, 2
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 $A$
2017 Saudi Arabia Pre-TST + Training Tests, 8
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.
2013 Tournament of Towns, 1
There are six points on the plane such that one can split them into two triples each creating a triangle. Is it always possible to split these points into two triples creating two triangles with no common point (neither inside, nor on the boundary)?
2001 Junior Balkan Team Selection Tests - Moldova, 1
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?
1969 Polish MO Finals, 6
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
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?
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?
2014 Finnish National High School Mathematics, 4
The radius $r$ of a circle with center at the origin is an odd integer.
There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers.
Determine $r$.
1978 Chisinau City MO, 158
Five points are selected on the plane so that no three of them lie on one straight line. Prove that some four of these five points are the vertices of a convex quadrilateral.
2014 Junior Balkan Team Selection Tests - Romania, 3
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 $2$.
1949 Moscow Mathematical Olympiad, 164
There are $12$ points on a circle. Four checkers, one red, one yellow, one green and one blue sit at neighboring points. In one move any checker can be moved four points to the left or right, onto the fifth point, if it is empty. If after several moves the checkers appear again at the four original points, how might their order have changed?