The tracks in a zoo form an equilateral triangle, in which the middle lines are drawn. A monkey ran away from its cage. Two guards try to catch the monkey. Will they be able to catch the monkey if all three of them can run only along the tracks, and the speed of the monkey and the speed of the guards are equal and they can always see each other?
The judges of an Olympiad decided to denote each participant with a natural number in such a way that it would be possible to unambiguously reconstruct the number of points received by each participant in each task, and that from each two participants the one with the greater number would be the participant which received a higher score. Help the judges solve this problem!
Given an endless piece of chequered paper with a cell side equal to one. The distance between two cells is the length of the shortest path parallel to cell lines from one cell to the other (it is considered the path of the center of a rook). What is the smallest number of colors to paint the board (each cell is painted with one color), so that two cells, located at a distance of 6, are always painted with different colors?