SIMON-MARAIS 2018 B3

Three spiders try to catch a beetle in a game. They are all initially positioned on the edges of a regular dodecahedron whose edges have length 1. At some point in time, they start moving continuously along the edges of the dodecahedron. The beetle and one of the spiders move with maximum speed 1, while the remaining two spiders move with maximum speed $\frac{1}{2018}$. Each player always knows their own position and the position of every other player. A player can turn around at any moment and can react to the behaviour of other players instantaneously. The spiders can communicate to decide on a strategy before and during the game. If any spider occupies the same position as the beetle at some time, then the spiders win the game.

Prove that the spiders can win the game, regardless of the initial positions of all players and regardless of how the beetle moves.

(A regular dodecahedron is a convex polyhedron with twelve faces, each of which is a pentagon with equal side lengths and equal angles. Three faces meet at each vertex.)