"Flawless Harmony" Puzzle

Source: AUSTMS Puzzle Corner 35

Problem:

Call a nine-digit number flawless if it has all the digits from 1 to 9 in some order. An unordered pair of flawless numbers is called harmonious if they sum to 987654321. Note that (a, b) and (b, a) are considered to be the same unordered pair.

Without resorting to an exhaustive search, prove that the number of harmonious pairs is odd.

Update (23 Oct 2014):
Solution: Posted by me (Pratik Poddar) in comments!

Minimum sum of numbers in an array

Source: Asked to me on quora ( cseblog.quora.com )

Problem:

Given an array of n positive numbers (n ~ 100000), what is the algorithmic approach to find the minimum possible sum (>=0) by using all the numbers in an array?

Example 1:
1 2 2 3 4
Answer : 0 (-1+2-2-3+4)

Example 2:
2 3 4 7 13
Answer: 1 (+2-3-4-7+13)

Caterer's Problem

Source: Puzzle Toad CMU

Problem:

You are organizing a conference, with a festive dinner on the first day. The catering service has 1024 different dinner choices they know how to make, out of which you need to choose 10 to be in the dinner menu (each participant will choose one of these during the dinner). You send an email to the 6875 participants of the conference, with the list of all 1024 choices, asking them to rank the choices in linear order from their favorite to their unfavorite.

You want to find a list L of 10 choices, such that for any dinner choice d not in the list L, if we run a vote of d against L, at least half of people will favor one of the choices in L over d (it may be different dish for different people).

Is it always possible to produce such a list?