This is harhrayr's solution.

let dp_N, diff — the maximum amount we can give to the first person, if we give him diff more than we give to the second person. Note that diff can be negative
For every number, we can give it to the first person, the second person, or noone.
So, dp_N, diff = max(dp_{N - 1, diff}, C_N + dp_{N - 1, diff - CN}, dp_{N - 1, diff + CN}). The first option corresponds to giving the banknote to noone, the second option is giving it to the first person, and the third option is giving it to the second person.
Then the answer is dp_N, 0

For the 4th problem, I first scaled all the numbers up to integers then for each number I find all factors of that number that would be valid.

Next, I do a bitmask dp with those factors. It seems as if this wouldn't run in time. However, because so many factors overlap and because only a limited number of factors could be valid, we do not hit that many states. In the case where we have a ton of valid factors, nearly all of those will just hit the same number and that greatly betters our runtime.

-EDIT- Informal Proof: A large factor will only change 1 or 2 bits by satisfying only one or two numbers where a smaller factor will satisfy a ton of numbers because it must hit all those in the range [a, b].

Furthermore, many factors are divisors of other factors. When we choose a factor, any factor that it divides will thus never be chosen later on because it will be useless. This greatly limits the number of states we hit and will allow us to have a reasonable runtime.

Of course, the proof for the actual upperbound is probably quite complicated and in contest I simply submitted on the intuition alone because the solution is very short and only took a few minutes to write.

The sequence is

khzyubiqkujupffyozvoxutbndrjxlkuixaipycgngpdtnnbblytqomnaazryn (the 1st string)
vkhzyubiqkujupffyozvoxutbndrjxlkuixaipycgngpdtnnbblytqomnaazryn (the 11th string)
ckhzyubiqkujupffyozvoxutbndrjxlkuixaipycgngpdtnnbblytqomnaazryn (the 7th string)
ockhzyubiqkujupffyozvoxutbndrjxlkuixaipycgngpdtnnbblytqomnaazryn (the 5th string)
aockhzyubiqkujupffyozvoxutbndrjxlkuixaipycgngpdtnnbblytqomnaazryn (the 2nd string)
haockhzyubiqkujupffyozvoxutbndrjxlkuixaipycgngpdtnnbblytqomnaazryn (the 15th string).

The sequence don't need to be a subsequence.