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

2022 239 Open Mathematical Olympiad, 3
Tags: color , geometry , combinatorial geometry , point
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.

2016 Saint Petersburg Mathematical Olympiad, 4
Tags: combinatorial geometry , combinatorics , speed , circles , point
$N> 4$ points move around the circle, each with a constant speed. For Any four of them have a moment in time when they all meet. Prove that is the moment when all the points meet.

1978 Dutch Mathematical Olympiad, 3
Tags: combinatorics , combinatorial geometry , point
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).

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

1999 Singapore MO Open, 1
Tags: even , coloring , combinatorics , point
Let $n$ be a positive integer. A square $ABCD$ is divided into $n^2$ identical small squares by drawing $(n-1)$ equally spaced lines parallel to the side $AB$ and another $(n- 1)$ equally spaced lines parallel to $BC$, thus giving rise to $(n+1)^2$ intersection points. The points $A, C$ are coloured red and the points $B, D$ are coloured blue. The rest of the intersection points are coloured either red or blue. Prove that the number of small squares having exactly $3$ vertices of the same colour is even.

1982 Czech and Slovak Olympiad III A, 4
Tags: geometry , combinatorics , combinatorial geometry , point
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 $\frac12$.

2018 Iran MO (1st Round), 4
Tags: point
There are $5$ points in the plane no three of which are collinear. We draw all the segments whose vertices are these points. What is the minimum number of new points made by the intersection of the drawn segments? $\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 5$

1998 All-Russian Olympiad Regional Round, 10.3
Tags: combinatorial geometry , geometry , point , combinatorics
Prove that from any finite set of points on the plane, you can remove a point from the bottom in such a way that the remaining set can be split into two parts of smaller diameter. (Diameter is the maximum distance between points of the set.) [hide=original wording]Докажите, что из любого конечного множества точек на плоскости можно так удалитьо дну точку, что оставшееся множество можно разбить на две части меньшего диаметра. (Диаметр—это максимальное расстояние между точками множества.)[/hide]

2015 Ukraine Team Selection Test, 9
Tags: heptagon , pentagon , convex , combinatorics , combinatorial geometry , point
The set $M$ consists of $n$ points on the plane and satisfies the conditions: $\bullet$ there are $7$ points in the set $M$, which are vertices of a convex heptagon, $\bullet$ for arbitrary five points with $M$, which are vertices of a convex pentagon, there is a point that also belongs to $M$ and lies inside this pentagon. Find the smallest possible value that $n$ can take .

1981 Poland - Second Round, 5
Tags: combinatorics , geometry , combinatorial geometry , point
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 $.

1999 North Macedonia National Olympiad, 4
Tags: combinatorial geometry , line , point
Do there exist $100$ straight lines on a plane such that they intersect each other in exactly $1999$ points?

2018 Estonia Team Selection Test, 11
Tags: combinatorics , combinatorial geometry , point
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 $Y$

1978 Chisinau City MO, 163
Tags: polyline , geometry , combinatorial geometry , geometric inequality , point
On the plane $n$ points are selected that do not belong to one straight line. Prove that the shortest closed path passing through all these points is a non-self-intersecting polygon.

2012 Tournament of Towns, 2
Tags: combinatorics , combinatorial geometry , circles , point
One hundred points are marked inside a circle, with no three in a line. Prove that it is possible to connect the points in pairs such that all fifty lines intersect one another inside the circle.

2003 Junior Balkan Team Selection Tests - Romania, 4
Tags: coloring , combinatorial geometry , point , combinatorics
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.

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

2009 Moldova National Olympiad, 7.3
Tags: geometry , point
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.

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

2000 BAMO, 4
Tags: combinatorics , combinatorial geometry , distance , point
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.

2010 Contests, 1
Tags: combinatorics , combinatorial geometry , collinear , point
$3n$ 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.

2014 IFYM, Sozopol, 2
Tags: number theory , point , circles , prime
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 $r$.

1955 Moscow Mathematical Olympiad, 318
Tags: maximum , combinatorics , combinatorial geometry , point
What greatest number of triples of points can be selected from $1955$ given points, so that each two triples have one common point?

1998 German National Olympiad, 1
Tags: combinatorics , combinatorial geometry , line , point
Find all possible numbers of lines in a plane which intersect in exactly $37$ points.

1986 Tournament Of Towns, (131) 7
Tags: combinatorics , combinatorial geometry , arc , angle , circles , point
On the circumference of a circle are $21$ points. Prove that among the arcs which join any two of these points, at least $100$ of them must subtend an angle at the centre of the circle not exceeding $120^o$ . ( A . F . Sidorenko)

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