ILL 1987 Problem 1

Let $x_1, x_2,\cdots, x_n$ be $n$ integers. Let $n = p + q$ , where $p$ and $q$ are positive integers. For $i = 1, 2, \cdots, n$ , put $$S_i = x_i + x_{i+1} +\cdots + x_{i+p-1} \text{ and } T_i = x_{i+p} + x_{i+p+1} +\cdots + x_{i+n-1}$$ (it is assumed that $x_{i+n }= x_i$ for all $i$ ). Next, let $m(a, b)$ be the number of indices $i$ for which $S_i$ leaves the remainder $a$ and $T_i$ leaves the remainder $b$ on division by $3$ , where $a, b \in \{0, 1, 2\}$ . Show that $m(1, 2)$ and $m(2, 1)$ leave the same remainder when divided by $3.$