Found problems: 121
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?
1991 Denmark MO - Mohr Contest, 5
Show that no matter how $15$ points are plotted within a circle of radius $2$ (circle border included), there will be a circle with radius $1$ (circle border including) which contains at least three of the $15$ 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 $n$.
Estonia Open Junior - geometry, 2014.2.5
In the plane there are six different points $A, B, C, D, E, F$ such that $ABCD$ and $CDEF$ are parallelograms. What is the maximum number of those points that can be located on one circle?
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.
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)
1961 Kurschak Competition, 1
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 $\sqrt2$.
2015 Finnish National High School Mathematics Comp, 5
Mikko takes a multiple choice test with ten questions. His only goal is to pass the test, and this requires seven points. A correct answer is worth one point, and answering wrong results in the deduction of one point. Mikko knows for sure that he knows the correct answer in the six first questions. For the rest, he estimates that he can give the correct answer to each problem with probability $p, 0 < p < 1$. How many questions Mikko should try?
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.
2015 JBMO Shortlist, C2
$2015$ 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.
2014 IFYM, Sozopol, 2
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 $r$.
2014 NZMOC Camp Selection Problems, 4
Given $2014$ points in the plane, no three of which are collinear, what is the minimum number of line segments that can be drawn connecting pairs of points in such a way that adding a single additional line segment of the same sort will always produce a triangle of three connected 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 $1.5$.
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.
1987 Polish MO Finals, 1
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 $P_n$.
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.
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 $4$)
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 $QR$.
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.