IrMO 2014 Problem 10

Over a period of $k$ consecutive days, a total of $2014$ babies were born in a certain city, with at least one baby being born each day. Show that: (a) If $1014 < k \le 2014$ , there must be a period of consecutive days during which exactly $100$ babies were born. (b) By contrast, if $k = 1014$ , such a period might not exist.