11st ibmo - costa rica 1996/q5.
Source: Spanish Communities
April 23, 2006
combinatorics unsolvedcombinatorics
Problem Statement
Three tokens , , are, each one in a vertex of an equilateral triangle of side . Its divided on equilateral triangles of side 1, such as it is shown in the figure for the case Initially, all the lines of the figure are painted blue. The tokens are moving along the lines painting them of red, following the next two rules:(1) First moves, after that moves, and then , by turns. On each turn, the token moves over exactly one line of one of the little triangles, form one side to the other.
(2) Non token moves over a line that is already painted red, but it can rest on one endpoint of a side that is already red, even if there is another token there waiting its turn.Show that for every positive integer it is possible to paint red all the sides of the little triangles.