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: 81

2021 European Mathematical Cup, 4
Tags: emc , European Mathematical Cup , combinatorics , partitions , Coloring , Ramsey Theory
Let $n$ be a positive integer. Morgane has coloured the integers $1,2,\ldots,n$. Each of them is coloured in exactly one colour. It turned out that for all positive integers $a$ and $b$ such that $a<b$ and $a+b \leqslant n$, at least two of the integers among $a$, $b$ and $a+b$ are of the same colour. Prove that there exists a colour that has been used for at least $2n/5$ integers. \\ \\ (Vincent Jugé)

1984 IMO Shortlist, 8
Tags: geometry , combinatorial geometry , Coloring , Ramsey Theory , IMO , IMO 1984
Given points $O$ and $A$ in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point $X$ in the plane, the circle $C(X)$ has center $O$ and radius $OX+{\angle AOX\over OX}$, where $\angle AOX$ is measured in radians in the range $[0,2\pi)$. Prove that we can find a point $X$, not on $OA$, such that its color appears on the circumference of the circle $C(X)$.

2017 ITAMO, 5
Tags: number theory , algebra , Ramsey Theory , combinatorics
Let $ x_1 , x_2, x_3 ...$ a succession of positive integers such that for every couple of positive integers $(m,n)$ we have $ x_{mn} \neq x_{m(n+1)}$ . Prove that there exists a positive integer $i$ such that $x_i \ge 2017 $.

1985 IMO Longlists, 41
Tags: combinatorics , point set , combinatorial geometry , Ramsey Theory , IMO Shortlist
A set of $1985$ points is distributed around the circumference of a circle and each of the points is marked with $1$ or $-1$. A point is called “good” if the partial sums that can be formed by starting at that point and proceeding around the circle for any distance in either direction are all strictly positive. Show that if the number of points marked with $-1$ is less than $662$, there must be at least one good point.

1999 IMO Shortlist, 4
Tags: algebra , Coloring , partition , Ramsey Theory , IMO Shortlist , combinatorics
Prove that the set of positive integers cannot be partitioned into three nonempty subsets such that, for any two integers $x,y$ taken from two different subsets, the number $x^2-xy+y^2$ belongs to the third subset.

1987 IMO Shortlist, 18
Tags: combinatorics , partition , Coloring , Ramsey Theory , Set systems , IMO Shortlist
For any integer $r \geq 1$, determine the smallest integer $h(r) \geq 1$ such that for any partition of the set $\{1, 2, \cdots, h(r)\}$ into $r$ classes, there are integers $a \geq 0 \ ; 1 \leq x \leq y$, such that $a + x, a + y, a + x + y$ belong to the same class. [i]Proposed by Romania[/i]

1992 IMO Longlists, 77
Tags: combinatorics , Ramsey Theory , Additive Number Theory , Additive combinatorics , IMO Shortlist , Extremal combinatorics , IMO Longlist
Show that if $994$ integers are chosen from $1, 2,\cdots , 1992$ and one of the chosen integers is less than $64$, then there exist two among the chosen integers such that one of them is a factor of the other.

1992 IMO Longlists, 25
Tags: partition , Coloring , combinatorics , Ramsey Theory , IMO Shortlist , IMO Longlist
[b][i](a) [/i][/b] Show that the set $\mathbb N$ of all positive integers can be partitioned into three disjoint subsets $A, B$, and $C$ satisfying the following conditions: \[A^2 = A, B^2 = C, C^2 = B,\] \[AB = B, AC = C, BC = A,\] where $HK$ stands for $\{hk | h \in H, k \in K\}$ for any two subsets $H, K$ of $\mathbb N$, and $H^2$ denotes $HH.$ [b][i](b)[/i][/b] Show that for every such partition of $\mathbb N$, $\min\{n \in N | n \in A \text{ and } n + 1 \in A\}$ is less than or equal to $77.$

1986 IMO Longlists, 36
Tags: graph theory , combinatorics , IMO , Coloring , Ramsey Theory , Extremal combinatorics , IMO 1986
Given a finite set of points in the plane, each with integer coordinates, is it always possible to color the points red or white so that for any straight line $L$ parallel to one of the coordinate axes the difference (in absolute value) between the numbers of white and red points on $L$ is not greater than $1$?

2004 China Team Selection Test, 2
Tags: matrix , combinatorics , IMO , Ramsey Theory , Extremal combinatorics , IMO 2001 , IMO Shortlist
Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.

2017 Germany Team Selection Test, 2
Tags: combinatorics , IMO Shortlist , Coloring , Ramsey Theory
Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

2004 Bulgaria Team Selection Test, 2
Tags: combinatorics proposed , combinatorics , Ramsey Theory , graph theory
The edges of a graph with $2n$ vertices ($n \ge 4$) are colored in blue and red such that there is no blue triangle and there is no red complete subgraph with $n$ vertices. Find the least possible number of blue edges.

1978 IMO Shortlist, 10
Tags: graph theory , Ramsey Theory , combinatorics , Extremal Graph Theory , Extremal combinatorics , IMO , IMO 1978
An international society has its members from six different countries. The list of members contain $1978$ names, numbered $1, 2, \dots, 1978$. Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country.

1988 IMO Longlists, 54
Tags: combinatorics , partition , Coloring , Extremal combinatorics , Ramsey Theory , IMO Shortlist
Find the least natural number $ n$ such that, if the set $ \{1,2, \ldots, n\}$ is arbitrarily divided into two non-intersecting subsets, then one of the subsets contains 3 distinct numbers such that the product of two of them equals the third.

2017 Brazil Team Selection Test, 1
Tags: combinatorics , IMO Shortlist , Coloring , Ramsey Theory
Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

1985 IMO Shortlist, 14
Tags: combinatorics , point set , combinatorial geometry , Ramsey Theory , IMO Shortlist
A set of $1985$ points is distributed around the circumference of a circle and each of the points is marked with $1$ or $-1$. A point is called “good” if the partial sums that can be formed by starting at that point and proceeding around the circle for any distance in either direction are all strictly positive. Show that if the number of points marked with $-1$ is less than $662$, there must be at least one good point.

2017 Germany Team Selection Test, 2
Tags: combinatorics , IMO Shortlist , Coloring , Ramsey Theory
Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

2017 Romania Team Selection Test, P3
Tags: combinatorics , IMO Shortlist , Coloring , Ramsey Theory
Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

2001 IMO Shortlist, 8
Tags: matrix , combinatorics , IMO , Ramsey Theory , Extremal combinatorics , IMO 2001 , IMO Shortlist
Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.

1964 IMO Shortlist, 4
Tags: combinatorics , Ramsey Theory , Coloring , graph theory , IMO , IMO 1964 , Extremal combinatorics
Seventeen people correspond by mail with one another-each one with all the rest. In their letters only three different topics are discussed. each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.

2004 Bulgaria Team Selection Test, 2
Tags: combinatorics proposed , combinatorics , Ramsey Theory , graph theory
The edges of a graph with $2n$ vertices ($n \ge 4$) are colored in blue and red such that there is no blue triangle and there is no red complete subgraph with $n$ vertices. Find the least possible number of blue edges.

1983 IMO Shortlist, 1
Tags: combinatorics , Extremal combinatorics , Ramsey Theory , graph theory , Extremal Graph Theory , IMO Shortlist
The localities $P_1, P_2, \dots, P_{1983}$ are served by ten international airlines $A_1,A_2, \dots , A_{10}$. It is noticed that there is direct service (without stops) between any two of these localities and that all airline schedules offer round-trip flights. Prove that at least one of the airlines can offer a round trip with an odd number of landings.

1998 Turkey MO (2nd round), 3
Tags: Ramsey Theory , combinatorics proposed , combinatorics
The points of a circle are colored by three colors. Prove that there exist infinitely many isosceles triangles inscribed in the circle whose vertices are of the same color.

2017 Azerbaijan Team Selection Test, 3
Tags: combinatorics , IMO Shortlist , Coloring , Ramsey Theory
Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

2010 Romania National Olympiad, 3
Tags: Ramsey Theory , combinatorics unsolved , combinatorics
In the plane are given $100$ points, such that no three of them are on the same line. The points are arranged in $10$ groups, any group containing at least $3$ points. Any two points in the same group are joined by a segment. a) Determine which of the possible arrangements in $10$ such groups is the one giving the minimal numbers of triangles. b) Prove that there exists an arrangement in such groups where each segment can be coloured with one of three given colours and no triangle has all edges of the same colour. [i]Vasile Pop[/i]