Question 155
Let $S = \{1, \dots, 100\}$, and for every positive integer $n$ define
$$
T_n = \{(a_1, \dots, a_n) \in S^n \mid a_1 + \dots + a_n \equiv 0 \pmod{100}\}.
$$
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.
$$
T_n = \{(a_1, \dots, a_n) \in S^n \mid a_1 + \dots + a_n \equiv 0 \pmod{100}\}.
$$
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.