Olympiad problems

1983 IMO Shortlist, 14

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?

1986 IMO, 3

Tags: combinatorics , graph theory , coloring , ramsey theory , extremal combinatorics

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 $1$?

2011 Romania Team Selection Test, 2

Tags: number theory , floor function , pigeonhole principle , arithmetic sequence , ramsey 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]

2004 Bulgaria Team Selection Test, 2

Tags: ramsey theory , graph theory , combinatorics

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.

2004 Bulgaria Team Selection Test, 2

Tags: graph theory , combinatorics , ramsey 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.

1997 South africa National Olympiad, 6

Tags: graph theory , combinatorics , floor function , pigeonhole principle , ramsey theory

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.)