101 positive integers placed on a circle

Start at a certain position and form sums of subsequences of length $1, 2, \dotsc, 101$ starting at that position and going in clockwise direction. This is an increasing sequence of $101$ numbers so there are two different entries that are equal $\bmod$ $100$ (end in the same two digits). The difference between those entries is a positive multiple of $100$ and less than $300$ so either $100$ or $200$. This difference corresponds to a subsequence of numbers on the circle with sum either $100$ or $200$. If it is $200$ we're done, otherwise take the complement of that sequence.