Let $m$ be a positive integer. We say that a sequence of positive integers written on a circle is good , if the sum of any $m$ consecutive numbers on this circle is a power of $m$. 1. Let $n \geq 2$ be a positive integer. Prove that for any good sequence with $mn$ numbers, we can remove $m$ numbers such that the remaining $mn-m$ numbers form a good sequence. 2. Prove that in any good sequence with $m^2$ numbers, we can always find a number that was repeated at least $m$ times in the sequence.