KMO 2012 Problem 5

$p >3$ is a prime number such that $p | 2^{p-1} -1$ and $p \not | 2^x - 1$ for $x = 1, 2, \cdots , p-2$ . Let $p = 2k+3$ . Now we define sequence $\{ a_n \}$ as $$a_i = a_{i+k}= 2^i ( 1 \le i \le k ) , \ a_{j+2k} = a_j a_{j+k} \ ( j \ge 1 )$$ Prove that there exist $2k$ consecutive terms of sequence $a_{x+1} , a_{x+2} , \cdots , a_{x+2k}$ such that for all $1 \le i < j \le 2k$ , $a_{x+i} \not \equiv a_{x+j} \ (mod \ p)$ .