IrMO 1988 Problem 2

Let $x_1, . . . , x_n$ be $n$ integers, and let $p$ be a positive integer, with $p < n$ . Put $$S_1 = x_1 + x_2 + . . . + x_p$$ $$T_1 = x_{p+1} + x_{p+2} + . . . + x_n$$ $$S_2 = x_2 + x_3 + . . . + x_{p+1}$$ $$T_2 = x_{p+2} + x_{p+3} + . . . + x_n + x_1$$ $$...$$ $$S_n=x_n+x_1+...+x_{p-1}$$ $$T_n=x_p+x_{p+1}+...+x_{n-1}$$ For $a = 0, 1, 2, 3$ , and $b = 0, 1, 2, 3$ , let $m(a, b)$ be the number of numbers $i$ , $1 \leq i \leq n$ , such that $S_i$ leaves remainder $a$ on division by $4$ and $T_i$ leaves remainder $b$ on division by $4$ . Show that $m(1, 3)$ and $m(3, 1)$ leave the same remainder when divided by $4$ if, and only if, $m(2, 2)$ is even.