Problem

Source: Iranian 3rd round Number Theory exam P2

Tags: modular arithmetic, number theory proposed, number theory



We say two sequence of natural numbers A=($a_1,...,a_n$) , B=($b_1,...,b_n$)are the exchange and we write $A\sim B$. if $503\vert a_i - b_i$ for all $1\leq i\leq n$. also for natural number $r$ : $A^r$ = ($a_1^r,a_2^r,...,a_n^r$). Prove that there are natural number $k,m$ such that : $i$)$250 \leq k $ $ii$)There are different permutations $\pi _1,...,\pi_k$ from {$1,2,3,...,502$} such that for $1\leq i \leq k-1$ we have $\pi _i^m\sim \pi _{i+1}$ (15 points)