Let $n$ be a positive even integer, and let $a_1,...,a_n$ and $w_1, ..., w_n$ be integers satisfying $\sum_{ k=1}^n a_k\equiv\sum_{ k=1}^n w_k =0 (mod n)$. A conjecture of Bialostocki states that there is a permutation $\sigma$ on { 1,...,n} such that $\sum_{ k=1}^n w_k a_{ \sigma(k)}=0 (mod n)$. In this paper we confirm the conjecture when $w_1,...,w_n$ form an arithmetic progression with even common difference.
Categories: math.CO math.NT