Given a subset of the naturals that contains ten numbers chosen randomly from 10,11,..,99 prove that there exist two disjoint subsets of , whose elements have equal sums.
The number of proper subsets of A is 2^10-2=1022.
The lowest value of the sum of elements of a proper subset is 10.
The highest value of the sum of elements of a proper subset is 99+98+97+96+95+94+93+92+91+90=945.
So, the number of different values for the sum of elements of a proper subset is 945-10+1=936.
Then by the Pigeon-hole Principle two proper subsets have the same value for the sum of elements. If these subsets are disjoint, we are done. If, however, these subsets have some common element(s), we may consider the subsets formed by deleting the common element(s), and they will satisfy the required conditions.