Question 35
Let $n, m$ be positive integers. A sequence of $mn$ positive integers written on a circle is called nice if the sum of any $m$ consecutive integers is a power of $m$. Show that
(1) for any nice sequence of $mn$ ($m \ge 2$) integers, one can delete $m$ consecutive integers so that the remaining sequence of $m(n-1)$ integers is nice.
(2) any nice sequence of $m^2$ integers contains an integer which is repeated at least $m$ times.
(Bayarmagnai Gombodorj)
(1) for any nice sequence of $mn$ ($m \ge 2$) integers, one can delete $m$ consecutive integers so that the remaining sequence of $m(n-1)$ integers is nice.
(2) any nice sequence of $m^2$ integers contains an integer which is repeated at least $m$ times.
(Bayarmagnai Gombodorj)