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: 181

2008 Brazil Team Selection Test, 4
Tags: combinatorics , combinatorial geometry , polygon , Extremal combinatorics , IMO Shortlist
Given is a convex polygon $ P$ with $ n$ vertices. Triangle whose vertices lie on vertices of $ P$ is called [i]good [/i] if all its sides are unit length. Prove that there are at most $ \frac {2n}{3}$ [i]good[/i] triangles. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]

2010 Balkan MO Shortlist, C3
Tags: induction , geometry , trigonometry , combinatorics proposed , combinatorics , Extremal combinatorics , marvio
A strip of width $w$ is the set of all points which lie on, or between, two parallel lines distance $w$ apart. Let $S$ be a set of $n$ ($n \ge 3$) points on the plane such that any three different points of $S$ can be covered by a strip of width $1$. Prove that $S$ can be covered by a strip of width $2$.

1966 IMO Shortlist, 45
Tags: combinatorics , maximization , Extremal combinatorics , Combinatorics of words , IMO Shortlist , IMO Longlist
An alphabet consists of $n$ letters. What is the maximal length of a word if we know that any two consecutive letters $a,b$ of the word are different and that the word cannot be reduced to a word of the kind $abab$ with $a\neq b$ by removing letters.

2010 Contests, 3
Tags: induction , geometry , trigonometry , combinatorics proposed , combinatorics , Extremal combinatorics , marvio
A strip of width $w$ is the set of all points which lie on, or between, two parallel lines distance $w$ apart. Let $S$ be a set of $n$ ($n \ge 3$) points on the plane such that any three different points of $S$ can be covered by a strip of width $1$. Prove that $S$ can be covered by a strip of width $2$.

2009 IMO Shortlist, 2
Tags: combinatorics , Extremal combinatorics , IMO Shortlist
For any integer $n\geq 2$, let $N(n)$ be the maxima number of triples $(a_i, b_i, c_i)$, $i=1, \ldots, N(n)$, consisting of nonnegative integers $a_i$, $b_i$ and $c_i$ such that the following two conditions are satisfied: [list][*] $a_i+b_i+c_i=n$ for all $i=1, \ldots, N(n)$, [*] If $i\neq j$ then $a_i\neq a_j$, $b_i\neq b_j$ and $c_i\neq c_j$[/list] Determine $N(n)$ for all $n\geq 2$. [i]Proposed by Dan Schwarz, Romania[/i]

2009 Belarus Team Selection Test, 2
Tags: geometry , IMO Shortlist , combinatorics , Extremal combinatorics , point set , pigenhole principle , bezout s identity
In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements. [i]Proposed by Jorge Tipe, Peru[/i]

1991 IMO Shortlist, 9
Tags: combinatorics , point set , Turan s theorem , Extremal Graph Theory , Extremal combinatorics , IMO Shortlist
In the plane we are given a set $ E$ of 1991 points, and certain pairs of these points are joined with a path. We suppose that for every point of $ E,$ there exist at least 1593 other points of $ E$ to which it is joined by a path. Show that there exist six points of $ E$ every pair of which are joined by a path. [i]Alternative version:[/i] Is it possible to find a set $ E$ of 1991 points in the plane and paths joining certain pairs of the points in $ E$ such that every point of $ E$ is joined with a path to at least 1592 other points of $ E,$ and in every subset of six points of $ E$ there exist at least two points that are not joined?

1969 IMO Shortlist, 60
Tags: Extremal combinatorics , point set , combinatorial geometry , geometry , IMO Shortlist , IMO Longlist
$(SWE 3)$ Find the natural number $n$ with the following properties: $(1)$ Let $S = \{P_1, P_2, \cdots\}$ be an arbitrary finite set of points in the plane, and $r_j$ the distance from $P_j$ to the origin $O.$ We assign to each $P_j$ the closed disk $D_j$ with center $P_j$ and radius $r_j$. Then some $n$ of these disks contain all points of $S.$ $(2)$ $n$ is the smallest integer with the above property.

1992 IMO Longlists, 40
Tags: geometry , 3D geometry , sphere , combinatorics , Extremal combinatorics , IMO Shortlist , IMO Longlist
The colonizers of a spherical planet have decided to build $N$ towns, each having area $1/1000$ of the total area of the planet. They also decided that any two points belonging to different towns will have different latitude and different longitude. What is the maximal value of $N$?

2005 Poland - Second Round, 3
Tags: inequalities , combinatorics proposed , combinatorics , graph theory , Extremal combinatorics
In space are given $n\ge 2$ points, no four of which are coplanar. Some of these points are connected by segments. Let $K$ be the number of segments $(K>1)$ and $T$ be the number of formed triangles. Prove that $9T^2<2K^3$.

2007 Germany Team Selection Test, 2
Tags: combinatorics , graph theory , Extremal combinatorics , Tournament graphs , IMO Shortlist
An $ (n, k) \minus{}$ tournament is a contest with $ n$ players held in $ k$ rounds such that: $ (i)$ Each player plays in each round, and every two players meet at most once. $ (ii)$ If player $ A$ meets player $ B$ in round $ i$, player $ C$ meets player $ D$ in round $ i$, and player $ A$ meets player $ C$ in round $ j$, then player $ B$ meets player $ D$ in round $ j$. Determine all pairs $ (n, k)$ for which there exists an $ (n, k) \minus{}$ tournament. [i]Proposed by Carlos di Fiore, Argentina[/i]

2018 Bosnia and Herzegovina EGMO TST, 4
Tags: combinatorics , maximization , Extremal combinatorics
It is given positive integer $n$. Let $a_1, a_2,..., a_n$ be positive integers with sum $2S$, $S \in \mathbb{N}$. Positive integer $k$ is called separator if you can pick $k$ different indices $i_1, i_2,...,i_k$ from set $\{1,2,...,n\}$ such that $a_{i_1}+a_{i_2}+...+a_{i_k}=S$. Find, in terms of $n$, maximum number of separators

2018 Romania Team Selection Tests, 2
Tags: combinatorics , IMO Shortlist , Strings , sorting , distance , Extremal combinatorics , radius
Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

1990 IMO Longlists, 78
Tags: graph theory , combinatorics , Extremal combinatorics , Extremal Graph Theory , IMO Shortlist
Ten localities are served by two international airlines such that there exists a direct service (without stops) between any two of these localities and all airline schedules offer round-trip service between the cities they serve. Prove that at least one of the airlines can offer two disjoint round trips each containing an odd number of landings.

Russian TST 2018, P1
Tags: combinatorics , IMO Shortlist , Strings , sorting , distance , Extremal combinatorics , radius
Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

2018 Peru IMO TST, 6
Tags: combinatorics , IMO Shortlist , Strings , sorting , distance , Extremal combinatorics , radius
Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

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.

1990 IMO Longlists, 11
Tags: combinatorics , Extremal combinatorics , Extremal Graph Theory , graph theory , vertex degree , IMO Shortlist , IMO Longlist
In a group of mathematicians, every mathematician has some friends (the relation of friend is reciprocal). Prove that there exists a mathematician, such that the average of the number of friends of all his friends is no less than the average of the number of friends of all these mathematicians.

2018 Brazil Team Selection Test, 2
Tags: combinatorics , IMO Shortlist , Strings , sorting , distance , Extremal combinatorics , radius
Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

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 $1$?

1978 IMO Shortlist, 1
Tags: pigeonhole principle , combinatorics , partition , Set systems , Extremal combinatorics , IMO Shortlist
The set $M = \{1, 2, . . . , 2n\}$ is partitioned into $k$ nonintersecting subsets $M_1,M_2, \dots, M_k,$ where $n \ge k^3 + k.$ Prove that there exist even numbers $2j_1, 2j_2, \dots, 2j_{k+1}$ in $M$ that are in one and the same subset $M_i$ $(1 \le i \le k)$ such that the numbers $2j_1 - 1, 2j_2 - 1, \dots, 2j_{k+1} - 1$ are also in one and the same subset $M_j (1 \le j \le k).$

1998 USAMO, 4
Tags: rectangle , floor function , induction , combinatorics proposed , combinatorics , Extremal combinatorics , Hi
A computer screen shows a $98 \times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minimum number of mouse clicks needed to make the chessboard all one color.

2006 Germany Team Selection Test, 3
Tags: combinatorics , Extremal combinatorics , Coloring , polygon , diagonals , IMO Shortlist
Suppose we have a $n$-gon. Some $n-3$ diagonals are coloured black and some other $n-3$ diagonals are coloured red (a side is not a diagonal), so that no two diagonals of the same colour can intersect strictly inside the polygon, although they can share a vertex. Find the maximum number of intersection points between diagonals coloured differently strictly inside the polygon, in terms of $n$. [i]Proposed by Alexander Ivanov, Bulgaria[/i]

2018 Bulgaria EGMO TST, 2
Tags: graph theory , combinatorics , Extremal combinatorics
A country has $100$ cities and $n$ airplane companies which take care of a total of $2018$ two-way direct flights between pairs of cities. There is a pair of cities such that one cannot reach one from the other with just one or two flights. What is the largest possible value of $n$ for which between any two cities there is a route (a sequence of flights) using only one of the airplane companies?

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.