All numbers between $1$ and $10^{20}$ ($10^{20}$ not included) are divided into two sets: one with numbers, whose sum of digits is odd and the other with remaining numbers(those, whose sum of digits is even). Prove that sums of 10th power of numbers in both sets are equal.
I've tried to construct pairs from both sets with equal sums but nothing came out of it. Do you have any idea?