Question 68
Find the smallest value of $k$ for which $2012$ bags each containing finite number of colored balls, no matter how the contents of bags are arranged, can be distributed into $k$ boxes so that for each box at least one of the following two conditions is held:
i. all bags of a box contain a ball of the same color
ii. each bag of a box contains a ball colored differently from all balls of all other bags of this box.
i. all bags of a box contain a ball of the same color
ii. each bag of a box contains a ball colored differently from all balls of all other bags of this box.