IMO 1987 LL AUS1

Let x1, x2, . . . , xn be n integers. Let n = p + q, where p and q

imolonglistmathematicsolympiad

IMO 1987 LL AUS1

Origin: AUS

Problem

Let x1, x2, . . . , xn be n integers. Let n = p + q, where p and q are positive integers. For i = 1, 2, . . ., n, put Si = xi + xi+1 + \cdot \cdot \cdot + xi+p−1 and Ti = xi+p + xi+p+1 + \cdot \cdot \cdot + xi+n−1 (it is assumed that xi+n = xi for all i). Next, let m(a, b) be the number of indices i for which Si leaves the remainder a and Ti 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.