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2011 Lusophon Mathematical Olympiad

Part of Lusophon Mathematical Olympiad

Subcontests

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Perfect Deltas

Consider a sequence of equilateral triangles TnT_{n} 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; j11. A triangle is called a delta if its vertex is at the top; for example, there are 1010 deltas in T3T_{3}. A delta is said to be perfect if the length of its side is even. How many perfect deltas are there in T20T_{20}?

2011 Lusophon Mathematical Olympiad - Problem 6

Let dd be a positive real number. The scorpion tries to catch the flea on a 10×1010\times 10 chessboard. The length of the side of each small square of the chessboard is 11. In this game, the flea and the scorpion move alternately. The flea is always on one of the 121121 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 dd. 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 dd, the scorpion has a strategy to prevent the flea's escape. For what values of dd can the flea escape? Justify your answer.
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