TSTST 2018 Problem 6

Let $S=\{1, \ldots, 100\}$, and for every positive integer $n$ define $$T_{n}=\left\{\left(a_{1}, \ldots, a_{n}\right) \in S^{n} \mid a_{1}+\cdots+a_{n} \equiv 0 (\bmod 100)\right\}$$ Determine which $n$ have the following property: if we color any 75 elements of $S$ red, then at least half of the $n$-tuples in $T_{n}$ have an even number of coordinates with red elements.