This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 121

Denmark (Mohr) - geometry, 1991.5
Tags: point , geometry , combinatorial geometry , combinatorics , circles
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.

1969 Swedish Mathematical Competition, 6
Tags: point , triangle , combinatorial geometry , combinatorics
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?

1978 Dutch Mathematical Olympiad, 3
Tags: point , combinatorics , combinatorial geometry
There are $1978$ points in the flat plane. Each point has a circular disk with that point as its center and the radius is the distance to a fixed point. Prove that there are five of these circular disks, which together cover all $1978$ points (circular disk means: the circle and its inner area).

2000 Switzerland Team Selection Test, 15
Tags: point , geometry , combinatorial geometry , geometric inequalities , combinatorics
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 $x_1^2 +x_2^2 +...+x_{2000}^2<9$ .

2016 IMAR Test, 2
Tags: combinatorial geometry , combinatorics , point , intersection , line , polygon
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?

Kyiv City MO Juniors Round2 2010+ geometry, 2019.7.31
Tags: point , geometry , construction , analytic geometry , coordinate geometry
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 $B$.

1950 Moscow Mathematical Olympiad, 184
Tags: point , combinatorics , combinatorial geometry
* 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?

1981 Brazil National Olympiad, 4
Tags: point , combinatorial geometry , graph , combinatorics
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)?

1978 Chisinau City MO, 158
Tags: point , combinatorial geometry , combinatorics , geometry
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.

2016 Singapore Senior Math Olympiad, 2
Tags: point , combinatorial geometry , line , min , combinatorics
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.

2003 Junior Balkan Team Selection Tests - Romania, 4
Tags: point , coloring , combinatorics , combinatorial geometry
Show that one can color all the points of a plane using only two colors such that no line segment has all points of the same color.

2011 Tournament of Towns, 7
Tags: point , combinatorial geometry , combinatorics , circles
$100$ 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.

2000 Miklós Schweitzer, 2
Tags: point
Let $n$ red and $n$ blue subarcs of a circle be given such that each red subarc intersects each blue subarc. Prove that there is a point which is covered by at least $n$ of the given (red or blue) subarcs.

1982 All Soviet Union Mathematical Olympiad, 333
Tags: point , diameter , combinatorics , combinatorial geometry
$3k$ 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.

1973 Chisinau City MO, 63
Tags: point , combinatorics , coloring
Each point in space is colored in one of four different colors. Prove that there is a segment $1$ cm long with endpoints of the same color.

2002 Junior Balkan Team Selection Tests - Moldova, 2
Tags: point , combinatorics , combinatorial geometry , line
$64$ 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.

1997 Poland - Second Round, 6
Tags: point , geometry , 3d geometry , combinatorial geometry , cube , distance , max
Let eight points be given in a unit cube. Prove that two of these points are on a distance not greater than $1$.

2013 Tournament of Towns, 1
Tags: point , combinatorics , combinatorial geometry
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)?

2016 Latvia Baltic Way TST, 12
Tags: point , geometry
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?

2022 Sharygin Geometry Olympiad, 8.7
Tags: point , combinatorics , combinatorial geometry , geometry
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?

2011 German National Olympiad, 4
Tags: point , angle , maximal , sum , set , geometry
There are two points $A$ and $B$ in the plane. a) Determine the set $M$ of all points $C$ in the plane for which $|AC|^2 +|BC|^2 = 2\cdot|AB|^2.$ b) Decide whether there is a point $C\in M$ such that $\angle ACB$ is maximal and if so, determine this angle.

1956 Moscow Mathematical Olympiad, 324
Tags: point , combinatorics , combinatorial geometry , distance , geometry , circles
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 $10$.

2003 Portugal MO, 3
Tags: combinatorial geometry , point , geometry
Raquel painted $650$ points in a circle with a radius of $16$ cm. Shows that there is a circular crown with $2$ cm of inner radius and $3$ cm of outer radius that contain at least $10$ of these points.

2012 Tournament of Towns, 2
Tags: point , combinatorial geometry , combinatorics
One hundred points are marked in the plane, with no three in a line. Is it always possible to connect the points in pairs such that all fi fty segments intersect one another?

2012 QEDMO 11th, 8
Tags: point , combinatorial geometry , combinatorics
Prove that there are $2012$ points in the plane, none of which are three on one straight line and in pairs have integer distances .