KMO 2014

Let $p>5$ be a prime. Suppose that there exist integer $k$ such that $k^2 + 5$ is divisible by $p$ . Prove that there exist two positive integers $m,n$ satisfying $p^2 = m^2 + 5n^2$ .

In an island there are $n$ castles, and each castle is in country $A$ or $B$ . There is one commander per castle, and each commander belongs to the same country as the castle he's initially in. There are some (two-way) roads between castles (there may be roads between castles of different countries), and call two castles adjacent if there is a road between them. Prove that the following two statements are equivalent: (1) If some commanders from country $B$ move to attack an adjacent castle in country $A$ , some commanders from country $A$ could appropriately move in defense to adjacent castles in country $A$ so that in every castle of country $A$ , the number of country $A$ 's commanders defending that castle is not less than the number of country $B$ 's commanders attacking that castle. (Each commander can defend or attack only one castle at a time.) (2) For any arbitrary set $X$ of castles in country $A$ , the number of country $A$ 's castles that are in $X$ or adjacent to at least one of the castle in $X$ is not less than the number of country $B$ 's castles that are adjacent to at least one of the castles in $X$ .