KMO 2002 Problem 6

Let $p_n$ be the $n^{\text{th}}$ prime counting from the smallest prime $2$ in increasing order. For example, $p_1=2, p_2=3, p_3 =5, \cdots$ (a) For a given $n \ge 10$ , let $r$ be the smallest integer satisfying $$2\le r \le n-2, n-r+1 < p_r$$ and define $N_s=(sp_1p_2\cdots p_{r-1})-1$ for $s=1,2,\ldots, p_r$ . Prove that there exists $j, 1\le j \le p_r$ , such that none of $p_1,p_2,\cdots, p_n$ divides $N_j$ .

(b) Using the result of (a), find all positive integers $m$ for which $$p_{m+1}^2 < p_1p_2\cdots p_m$$