RIOPLATENSE 2017 Problem 6

For each fixed positiver integer $n$ , $n\geq 4$ and $P$ an integer, let $(P)_n \in [1, n]$ be the smallest positive residue of $P$ modulo $n$ . Two sequences $a_1, a_2, \dots, a_k$ and $b_1, b_2, \dots, b_k$ with the terms in $[1, n]$ are defined as equivalent, if there is $t$ positive integer, gcd $(t,n)=1$ , such that the sequence $(ta_1)_n, \dots, (ta_k)_n$ is a permutation of $b_1, b_2, \dots, b_k$ . Let $\alpha$ a sequence of size $n$ and your terms are in $[1, n]$ , such that each term appears $h$ times in the sequence $\alpha$ and $2h\geq n$ . Show that $\alpha$ is equivalent to some sequence $\beta$ which contains a subsequence such that your size is(at most) equal to $h$ and your sum is exactly equal to $n$ .