CIIM 2017 Problem 4

Let $m, n$ be positive integers and $a_1,\dots , a_m, b_1, \dots , b_n$ positive real numbers such that for every positive integer $k$ we have that $$(a_1^k + \cdots + a^k_m) - (b^k_1 + \cdots + b^k_n) \leq CkN,$$ for some fix $C$ and $N$ . Show that there exists $l \leq m, n$ and permutations $\sigma$ of $\{1, \dots , m\}$ and $\tau$ of $\{1,\dots , n\}$ , such that

1. $a\sigma(i) = b\tau(i)$ for $1 \leq i \leq l,$ 2. $a\sigma(i) , b\tau(i) \leq 1$ for $i > l.$