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 Francophone Mathematical Olympiad, 3
Tags: combinatorial geometry , combinatorics , Coloring , Francophone , Ramsey Theory
Every point in the plane was colored in red or blue. Prove that one the two following statements is true: $\bullet$ there exist two red points at distance $1$ from each other; $\bullet$ there exist four blue points $B_1$, $B_2$, $B_3$, $B_4$ such that the points $B_i$ and $B_j$ are at distance $|i - j|$ from each other, for all integers $i $ and $j$ such as $1 \le i \le 4$ and $1 \le j \le 4$.

2008 Brazil Team Selection Test, 4
Tags: geometry , rectangle , combinatorics , Ramsey Theory , Coloring , IMO Shortlist , RIP mavropnevma
In the Cartesian coordinate plane define the strips $ S_n \equal{} \{(x,y)|n\le x < n \plus{} 1\}$, $ n\in\mathbb{Z}$ and color each strip black or white. Prove that any rectangle which is not a square can be placed in the plane so that its vertices have the same color. [b]IMO Shortlist 2007 Problem C5 as it appears in the official booklet:[/b] In the Cartesian coordinate plane define the strips $ S_n \equal{} \{(x,y)|n\le x < n \plus{} 1\}$ for every integer $ n.$ Assume each strip $ S_n$ is colored either red or blue, and let $ a$ and $ b$ be two distinct positive integers. Prove that there exists a rectangle with side length $ a$ and $ b$ such that its vertices have the same color. ([i]Edited by Orlando Döhring[/i]) [i]Author: Radu Gologan and Dan Schwarz, Romania[/i]

2001 IMO, 3
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.

1986 IMO, 3
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 $1$?

1964 IMO, 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.

1996 Tuymaada Olympiad, 3
Tags: combinatorics , combinatorial geometry , Coloring , Ramsey Theory
Nine points of the plane, located at the vertices of a regular nonagon, are pairwise connected by segments, each of which is colored either red or blue. It is known that in any triangle with vertices at the vertices of the nonagon at least one side is red. Prove that there are four points, any two of which are connected by red lines.

1983 IMO Shortlist, 14
Tags: arithmetic sequence , Extremal combinatorics , Ramsey Theory , Arithmetic Progression , IMO , IMO 1983 , combinatorics
Is it possible to choose $1983$ distinct positive integers, all less than or equal to $10^5$, no three of which are consecutive terms of an arithmetic progression?

2007 IMO Shortlist, 5
Tags: geometry , rectangle , combinatorics , Ramsey Theory , Coloring , IMO Shortlist , RIP mavropnevma
In the Cartesian coordinate plane define the strips $ S_n \equal{} \{(x,y)|n\le x < n \plus{} 1\}$, $ n\in\mathbb{Z}$ and color each strip black or white. Prove that any rectangle which is not a square can be placed in the plane so that its vertices have the same color. [b]IMO Shortlist 2007 Problem C5 as it appears in the official booklet:[/b] In the Cartesian coordinate plane define the strips $ S_n \equal{} \{(x,y)|n\le x < n \plus{} 1\}$ for every integer $ n.$ Assume each strip $ S_n$ is colored either red or blue, and let $ a$ and $ b$ be two distinct positive integers. Prove that there exists a rectangle with side length $ a$ and $ b$ such that its vertices have the same color. ([i]Edited by Orlando Döhring[/i]) [i]Author: Radu Gologan and Dan Schwarz, Romania[/i]

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.

1983 IMO Longlists, 6
Tags: geometry , Ramsey Theory , combinatorics , Extremal combinatorics , right triangle , IMO , IMO 1983
Let $ABC$ be an equilateral triangle and $\mathcal{E}$ the set of all points contained in the three segments $AB$, $BC$, and $CA$ (including $A$, $B$, and $C$). Determine whether, for every partition of $\mathcal{E}$ into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle.

1997 South africa National Olympiad, 6
Tags: floor function , pigeonhole principle , graph theory , Ramsey Theory , combinatorics unsolved , combinatorics
Six points are connected in pairs by lines, each of which is either red or blue. Every pair of points is joined. Determine whether there must be a closed path having four sides all of the same colour. (A path is closed if it begins and ends at the same point.)

1978 USAMO, 5
Tags: AMC , USA(J)MO , USAMO , pigeonhole principle , floor function , Ramsey Theory
Nine mathematicians meet at an international conference and discover that among any three of them, at least two speak a common language. If each of the mathematicians speak at most three languages, prove that there are at least three of the mathematicians who can speak the same language.

1992 IMO Longlists, 10
Tags: combinatorics , Ramsey Theory , graph theory , IMO , IMO 1992 , Coloring , Extremal combinatorics
Consider $9$ points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of $\,n\,$ such that whenever exactly $\,n\,$ edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.

1990 IMO Longlists, 9
Tags: number theory , partition , Ramsey Theory , Coloring , Extremal combinatorics , IMO Shortlist , Hi
Assume that the set of all positive integers is decomposed into $ r$ (disjoint) subsets $ A_1 \cup A_2 \cup \ldots \cup A_r \equal{} \mathbb{N}.$ Prove that one of them, say $ A_i,$ has the following property: There exists a positive $ m$ such that for any $ k$ one can find numbers $ a_1, a_2, \ldots, a_k$ in $ A_i$ with $ 0 < a_{j \plus{} 1} \minus{} a_j \leq m,$ $ (1 \leq j \leq k \minus{} 1)$.

1986 IMO Shortlist, 9
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 $1$?

1983 IMO, 1
Tags: geometry , Ramsey Theory , combinatorics , Extremal combinatorics , right triangle , IMO , IMO 1983
Let $ABC$ be an equilateral triangle and $\mathcal{E}$ the set of all points contained in the three segments $AB$, $BC$, and $CA$ (including $A$, $B$, and $C$). Determine whether, for every partition of $\mathcal{E}$ into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle.

Russian TST 2017, P2
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.

1987 IMO Longlists, 56
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]

1983 IMO, 2
Tags: arithmetic sequence , Extremal combinatorics , Ramsey Theory , Arithmetic Progression , IMO , IMO 1983 , combinatorics
Is it possible to choose $1983$ distinct positive integers, all less than or equal to $10^5$, no three of which are consecutive terms of an arithmetic progression?

1988 IMO Shortlist, 20
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.

1984 IMO Longlists, 47
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 $C(X)$.

2008 South East Mathematical Olympiad, 1
Tags: combinatorics unsolved , combinatorics , Set systems , Additive combinatorics , Ramsey Theory
Given a set $S=\{1,2,3,\ldots,3n\},(n\in N^*)$, let $T$ be a subset of $S$, such that for any $x, y, z\in T$ (not necessarily distinct) we have $x+y+z\not \in T$. Find the maximum number of elements $T$ can have.

2011 Romania Team Selection Test, 2
Tags: floor function , pigeonhole principle , arithmetic sequence , Ramsey Theory , number theory proposed , number theory
Prove that the set $S=\{\lfloor n\pi\rfloor \mid n=0,1,2,3,\ldots\}$ contains arithmetic progressions of any finite length, but no infinite arithmetic progressions. [i]Vasile Pop[/i]

2017 India IMO Training Camp, 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.

2021 Cyprus JBMO TST, 4
Tags: combinatorics , pigeonhole principle , Ramsey Theory
We colour every square of a $4\times 19$ chess board with one of the colours red, green and blue. Prove that however this colouring is done, we can always find two horizontal rows and two vertical columns such that the $4$ squares on the intersections of these lines all have the same colour.