$100$ boxes of fruits - pick $51$ and get at least half of each type?

Friend told me this one, I'm completely stuck but also completely fascinated:

There are $100$ boxes with apples, oranges and bananas (mixed). How to Prove that you can pick $51$ boxes and to get at least half of all apples, at least half of all oranges and at least half of all bananas?

Edit: You can take a look in the boxes.

Solution 1:

This is apparently a (hard) problem from the Russian Math Olympiad which no one in the exam solved.

See here for a list of questions in that exam: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=32171

A solution for this problem is here: http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1367869#p1367869

A hint that was given (by Fedor Petrov):

If we have $2k$ boxes, we may partition them into two groups of $k$ boxes in such a way that number of apples in both groups differ by at most the maximal number of apples in a single box, and the same for oranges.