A pirate ship has 2015 treasure chests (all chests are closed). Each chest contains some amount of gold, some amount of silver and some amount of bronze. To distribute the gold, silver and bronze the pirates are going to do the following. The captain is going to decide first how many chests he wants to keep and tell that number to the rest of the pirates. Then he is going to open all the chests and decide which ones he wants to keep. The captain wants to make sure he can keep at least half of the total gold, half of the total silver and at least half of the total bronze. However, he wants to say the smallest possible number to keep the rest of the pirates as happy as he can. What number should the captain say? (The amount of gold, silver, bronze in each chest may be different)
NOTES:
$1.$ I've modified this problem from Example Problem 3.3.7 of the book 'Problems solving methods in combinatorics an approach to olympiad'(Pablo Soberon Bravo's book).
$2.$ Problem solved in the book for gold and silver. But I coludn't find anything for version of gold, silver, bronze.
