Olympiad problems

1990 IMO Shortlist, 4

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)$.

1978 IMO Longlists, 30

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.

1977 IMO Longlists, 32

Tags: Ramsey Theory , combinatorics proposed , combinatorics

In a room there are nine men. Among every three of them there are two mutually acquainted. Prove that some four of them are mutually acquainted.

2017 Taiwan TST Round 3, 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.

1992 IMO, 3

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.

2008 China Team Selection Test, 3

Tags: search , arithmetic sequence , combinatorics proposed , combinatorics , Ramsey Theory

Suppose that every positve integer has been given one of the colors red, blue,arbitrarily. Prove that there exists an infinite sequence of positive integers $ a_{1} < a_{2} < a_{3} < \cdots < a_{n} < \cdots,$ such that inifinite sequence of positive integers $ a_{1},\frac {a_{1} \plus{} a_{2}}{2},a_{2},\frac {a_{2} \plus{} a_{3}}{2},a_{3},\frac {a_{3} \plus{} a_{4}}{2},\cdots$ has the same color.