Found problems: 115
1981 Poland - Second Round, 5
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 $.
2016 Singapore Senior Math Olympiad, 2
Let $n$ be a positive integer. Determine the minimum number of lines that can be drawn on the plane so that they intersect in exactly $n$ distinct points.
1982 Czech and Slovak Olympiad III A, 4
In a circle with a radius of $1$, $64$ mutually different points are selected. Prove that $10$ mutually different points can be selected from them, which lie in a circle with a radius $\frac12$.
1965 Polish MO Finals, 3
$ n > 2 $ points are chosen on a circle and each of them is connected to every other by a segment. Is it possible to draw all of these segments in one sequence, i.e. so that the end of the first segment is the beginning of the second, the end of the second - the beginning of the third, etc., and so that the end of the last segment is the beginning of the first?
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.
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 $2$.
2009 IMAC Arhimede, 6
At a football tournament, each team plays with each of the remaining teams, winning three points for the win, one point for the draw score and zero points for the defeat. At the end of the tournament it turned out that the sum of the winning points of all teams was $50$.
(a) How many teams participated in this tournament?
(b) How big is the difference between the team with the highest number and the number of points won?
1986 China Team Selection Test, 4
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.
2012 QEDMO 11th, 8
Prove that there are $2012$ points in the plane, none of which are three on one straight line and in pairs have integer distances .
2010 IMAC Arhimede, 1
$3n$ points are given ($n\ge 1$) in the plane, each $3$ of them are not collinear. Prove that there are $n$ distinct triangles with the vertices those points.
Kyiv City MO Juniors Round2 2010+ geometry, 2019.7.31
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 $B$.
1979 All Soviet Union Mathematical Olympiad, 283
Given $n$ points (in sequence)$ A_1, A_2, ... , A_n$ on a line. All the segments $A_1A_2$, $A_2A_3$,$ ...$, $A_{n-1}A_n$ are shorter than $1$. We need to mark $(k-1)$ points so that the difference of every two segments, with the ends in the marked points, is shorter than $1$. Prove that it is possible
a) for $k=3$,
b) for every $k$ less than $(n-1)$.
1941 Moscow Mathematical Olympiad, 087
On a plane, several points are chosen so that a disc of radius $1$ can cover every $3$ of them. Prove that a disc of radius $1$ can cover all the points.
2008 Thailand Mathematical Olympiad, 10
On the sides of triangle $\vartriangle ABC$, $17$ points are added, so that there are $20$ points in total (including the vertices of $\vartriangle ABC$.) What is the maximum possible number of (nondegenerate) triangles that can be formed by these points.
2015 Dutch IMO TST, 1
Let $a$ and $b$ be two positive integers satifying $gcd(a, b) = 1$. Consider a pawn standing on the grid point $(x, y)$.
A step of type A consists of moving the pawn to one of the following grid points: $(x+a, y+a),(x+a,y-a), (x-a, y + a)$ or $(x - a, y - a)$.
A step of type B consists of moving the pawn to $(x + b,y + b),(x + b,y - b), (x - b,y + b)$ or $(x - b,y - b)$.
Now put a pawn on $(0, 0)$. You can make a ( nite) number of steps, alternatingly of type A and type B, starting with a step of type A. You can make an even or odd number of steps, i.e., the last step could be of either type A or type B.
Determine the set of all grid points $(x,y)$ that you can reach with such a series of steps.
1990 All Soviet Union Mathematical Olympiad, 513
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.
2002 Junior Balkan Team Selection Tests - Moldova, 2
$64$ distinct points are positioned in the plane so that they determine exactly $2003$ different lines. Prove that among the $64$ points there are at least $4$ collinear points.
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.
2020 Ukrainian Geometry Olympiad - April, 5
On the plane painted $101$ points in brown and another $101$ points in green so that there are no three lying on one line. It turns out that the sum of the lengths of all $5050$ segments with brown ends equals the length of all $5050$ segments with green ends equal to $1$, and the sum of the lengths of all $10201$ segments with multicolored equals $400$. Prove that it is possible to draw a straight line so that all brown points are on one side relative to it and all green points are on the other.
2015 Peru MO (ONEM), 1
If $C$ is a set of $n$ points in the plane that has the following property: For each point $P$ of $C$, there are four points of $C$, each one distinct from $P$ , which are the vertices of a square. Find the smallest possible value of $n$.
2022 239 Open Mathematical Olympiad, 3
Let $A$ be a countable set, some of its countable subsets are selected such that; the intersection of any two selected subsets has at most one element. Find the smallest $k$ for which one can ensure that we can color elements of $A$ with $k$ colors such that each selected subsets exactly contain one element of one of the colors and an infinite number of elements of each of the other colors.
2015 Dutch IMO TST, 1
Let $a$ and $b$ be two positive integers satifying $gcd(a, b) = 1$. Consider a pawn standing on the grid point $(x, y)$.
A step of type A consists of moving the pawn to one of the following grid points: $(x+a, y+a),(x+a,y-a), (x-a, y + a)$ or $(x - a, y - a)$.
A step of type B consists of moving the pawn to $(x + b,y + b),(x + b,y - b), (x - b,y + b)$ or $(x - b,y - b)$.
Now put a pawn on $(0, 0)$. You can make a ( nite) number of steps, alternatingly of type A and type B, starting with a step of type A. You can make an even or odd number of steps, i.e., the last step could be of either type A or type B.
Determine the set of all grid points $(x,y)$ that you can reach with such a series of steps.
2010 Contests, 1
$3n$ points are given ($n\ge 1$) in the plane, each $3$ of them are not collinear. Prove that there are $n$ distinct triangles with the vertices those points.
2009 Sharygin Geometry Olympiad, 5
Let $n$ points lie on the circle. Exactly half of triangles formed by these points are acute-angled. Find all possible $n$.
(B.Frenkin)
2009 All-Russian Olympiad Regional Round, 11.5
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.