Kangaroos

From the Netherlands Junior Maths Olympiad:

There is a kangaroo in the centre of each square of a 5-by-5 grid of squares. Lightning strikes, and each kangaroo simultaneously jumps one square up, down, left or right. What is the greatest possible number of empty squares that could remain?

A Game of Stones

A PuzzleCritic Original:

There are 999 stones in a pile. Amisi and Boaz take it in turns removing either 3 or 5 stones from the pile, with Amisi going first, until no more moves are possible. The last player to make a move wins. Which player can guarantee victory?

L’s on a square

Happy New Year! From the 2004 Georg Mohr Contest:

Find all positive integers n such that a 2n x 2n chessboard can be covered by non-overlapping L-pieces, each covering 4 squares. Rotations and reflections are allowed.

42 Points

From the 2007 Australian Maths Competition:

There are 42 points $P_1, P_2, ..., P_{42}$ , placed in order on a straight line so that each distance from $P_i$ to $P_{i+1}$ is  $\dfrac{1}{i}$  for $1\leq i\leq 41$. What is the sum of the distances between every pair of these points?

Subsets of subsets

From the 1992 Irish Maths Olympiad:

Let A be a nonempty set with n elements. Find the number of ways of choosing a pair of subsets (B,C) of A such that B is a nonempty subset of C.