A number theory problem by Anik Mandal
.A fly is being chased by three spiders on the edges of a regular octahedron. The fly has a speed of meters per second, while each of the spiders has a speed of meters per second. The spiders choose the (distinct) starting positions of all the bugs, with the requirement that the fly must begin at a vertex. Each bug knows the position of each other bug at all times, and the goal of the spiders is for at least one of them to catch the fly. What is the maximum so that for any , the fly can always avoid being caught?
The answer is 25.
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1 solution
Consider the X as the fly and the dots as the spiders. The image above is the worst case scenario for the fly The two spiders on the vertices(they can be on any point on the side) chase down the fly so it has only 2 options side A and B. There is a spider on the midpoint as shown in the figure which will reach the fly by moving towards whichever direction the fly took. The spider must travel a distance half of that of the fly. In order for the fly to not be caught,it must travel twice as fast as the spider ie 2*r<50.Therefore r<25.