Problem: A rectangular table has 100 coins placed on it (centers must be on the table) such that none of the coins overlap, and it is impossible to place any more coins on the table without causing an overlap.
Prove that using 400 coins and allowing overlaps, we can cover the entire table.
1) Coins are identical and round.
2) You are given a configuration of 100 coins satisfying the first condition. You are to prove that there exists a fresh configuration of 400 coins satisfying the second condition.
3) Covering the table means for all points on the table, there is at least one coin above that point
4) Coins placed on the table means that the center of the coin is on the table
Solution posted by Stephen Rong in comments!