VTMO 2026 - Grade 8 - Semifinal - SET 10 - Self-study Question 5

Numbers $a_1,\ldots,a_n$ are each $1$ or $-1$. Put $S_n=a_1a_2+a_2a_3+\cdots+a_{n-1}a_n+a_na_1$. Can $S_n=0$ occur when (a) $n=30$; (b) $n=28$? Justify.