100 coins are placed on a rectangular table, such that no more coins can be added without overlapping. Show that you can cover the entire table with 400 coins (overlapping allowed).
Since we can not place any more coins on the table, each point of it is at distance at most 2r from the center of some coin, where r is the raidus of the coin. Now shrink the entire table twice in width and length, then replace every shrunk coin with a full sized one. This way the small table will be completely covered, because every point of it will be at distance at most r from the center of some coin. Add three more of these smaller tables, covered with coins, to create a covering of the big table.