Assume $A=\{a_1,\dots,a_{10}\}$, each $9<a_i<100$ ,then prove that there exists set B and C such that $A=B \cup C$ , $B \cap C = \emptyset$ and the elements' sum in B is equal to C's.
I am thinking that A has 1024 different subsets and the sum of each subset is no larger than 1000 which shows that there must be two sets B and C that their sums are equal, but I can't prove that they are disjoint and cover A.
Edit: This problem is itself wrong by discussion , let we look at another question : Use the same situation with the last problem, then will there exist two different partitions by two subsets such that they have the same sum on each subset?
Suppose the sum of $A$ is odd, then we can't partition $A$ into $B$ and $C$ such that they have the same sum.