3
Problems(2)
Perfect Deltas
Source: 2011 LMO Problem #3
9/21/2011
Consider a sequence of equilateral triangles as represented below:
[asy]
defaultpen(linewidth(0.8));size(350);
real r=sqrt(3);
path p=origin--(2,0)--(1,sqrt(3))--cycle;
int i,j,k;
for(i=1; i<5; i=i+1) {
for(j=0; j
combinatorics unsolvedcombinatorics
2011 Lusophon Mathematical Olympiad - Problem 6
Source: 2011 Lusophon Mathematical Olympiad - Problem 6
9/17/2011
Let be a positive real number. The scorpion tries to catch the flea on a chessboard. The length of the side of each small square of the chessboard is . In this game, the flea and the scorpion move alternately. The flea is always on one of the vertexes of the chessboard and, in each turn, can jump from the vertex where it is to one of the adjacent vertexes. The scorpion moves on the boundary line of the chessboard, and, in each turn, it can walk along any path of length less than .
At the beginning, the flea is at the center of the chessboard and the scorpion is at a point that he chooses on the boundary line. The flea is the first one to play. The flea is said to escape if it reaches a point of the boundary line, which the scorpion can't reach in the next turn. Obviously, for big values of , the scorpion has a strategy to prevent the flea's escape. For what values of can the flea escape? Justify your answer.
combinatorics proposedcombinatorics