KöMaL 2013 K367

A school organized an ice cream building competition for the students. The participants built their ice cream towers on 10-cm tall cones, by placing the scoops one by one on top of each other. Initially, each scoop was a spherical ball of 4-cm diameter, but they were compressed under the weight of the overlying balls. The height of a ball decreased by 1 mm owing to each ball on top of it. The prize winning ice cream tower was 47.5 cm tall, measured from the bottom of the cone to the top of the uppermost ball, and one third of the height of the lowermost ball was inside the cone. How many balls were used to build this tower?