RIOPLATENSE 2007 Problem 3

Let $p > 3$ be a prime number and $x$ an integer, denote by $r ( x )\in \{ 0 , 1 , ... , p - 1 \}$ to the rest of $x$ modulo $p$ . Let $x_1, x_2, ... , x_k$ ( $2 < k < p$ ) different integers modulo $p$ and not divisible by $p$ . We say that a number $a \in \{ 1 , 2 ,..., p -1 \}$ is good if $r ( a x_1) < r ( a x_2) <...< r ( a x_k)$ . Show that there are at most $\frac{2 p}{k + 1}-{ 1}$ good numbers.