Found problems: 115
2020 Federal Competition For Advanced Students, P2, 2
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)
1998 Belarusian National Olympiad, 7
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?
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.
1969 Swedish Mathematical Competition, 6
Given $3n$ points in the plane, no three collinear, is it always possible to form $n$ triangles (with vertices at the points), so that no point in the plane lies in more than one triangle?
Denmark (Mohr) - geometry, 1991.5
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.
1950 Moscow Mathematical Olympiad, 184
* On a circle, $20$ points are chosen. Ten non-intersecting chords without mutual endpoints connect some of the points chosen. How many distinct such arrangements are there?
2010 Sharygin Geometry Olympiad, 8
Given is a regular polygon. Volodya wants to mark $k$ points on its perimeter so that any another regular polygon (maybe having a different number of sides) doesn’t contain all marked points on its perimeter. Find the minimal $k$ sufficient for any given polygon.
2011 Tournament of Towns, 7
$100$ red points divide a blue circle into $100$ arcs such that their lengths are all positive integers from $1$ to $100$ in an arbitrary order. Prove that there exist two perpendicular chords with red endpoints.
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.
2018 Estonia Team Selection Test, 11
Let $k$ be a positive integer. Find all positive integers $n$, such that it is possible to mark $n$ points on the sides of a triangle (different from its vertices) and connect some of them with a line in such a way that the following conditions are satisfied:
1) there is at least $1$ marked point on each side,
2) for each pair of points $X$ and $Y$ marked on different sides, on the third side there exist exactly $k$ marked points which are connected to both $X$ and $Y$ and exactly k points which are connected to neither $X$ nor $Y$
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?
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 $\sqrt2$.
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.
2001 Tuymaada Olympiad, 1
$16$ chess players held a tournament among themselves: every two chess players played exactly one game. For victory in the party was given $1$ point, for a draw $0.5$ points, for defeat $0$ points. It turned out that exactly 15 chess players shared the first place. How many points could the sixteenth chess player score?
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 $n$.
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.
1955 Moscow Mathematical Olympiad, 318
What greatest number of triples of points can be selected from $1955$ given points, so that each two triples have one common point?
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?
2009 Moldova National Olympiad, 7.3
On the lines $AB$ are located $2009$ different points that do not belong to the segment $[AB]$. Prove that the sum of the distances from point $A$ to these points is not equal to the sum of the distances from point $B$ to these points.
2000 Switzerland Team Selection Test, 15
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 $x_1^2 +x_2^2 +...+x_{2000}^2<9$ .
1965 Dutch Mathematical Olympiad, 4
We consider a number of points in a plane. Each of these points is connected to at least one of the other points by a line segment, in such a way that a figure arises that does not break up into different parts (that is, from any point along drawn line segments we can reach any other point).. We assign a point the ”order” $n$, when in this point $n$ line segments meet. We characterize the obtained figure by writing down the order of each of its points one after the other. For example, a hexagon is characterized by the combination $\{2,2,2,2,2,2\}$ and a star with six rays by $\{6,1,1,1,1,1,1\}$.
(a) Sketch a figure' belonging to the combination $\{4,3,3,3,3\}$.
(b) Give the combinations of all possible figures, of which the sum of the order numbers is equal to $6$.
(c) Prove that every such combination contains an even number of odd numbers.
1962 Putnam, A1
Consider $5$ points in the plane, such that there are no $3$ of them collinear. Prove that there is a convex quadrilateral with vertices at $4$ points.
1972 Polish MO Finals, 2
On the plane are given $n > 2$ points, no three of which are collinear. Prove that among all closed polygonal lines passing through these points, any one with the minimum length is non-selfintersecting.
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.
2016 Latvia Baltic Way TST, 12
For what positive numbers $m$ and $n$ do there exist points $A_1, ..., Am$ and $B_1 ..., B_n$ in the plane such that, for any point $P$, the equation $$|PA_1|^2 +... + |PA_m|^2 =|PB_1|^2+...+|PA_n|^2 $$ holds true?