VMO 2013 Problem 4

Write down some numbers $a_1,a_2,\ldots, a_n$ from left to right on a line. Step 1, we write $a_1+a_2$ between $a_1,a_2$ ; $a_2+a_3$ between $a_2,a_3$ , …, $a_{n-1}+a_n$ between $a_{n-1},a_n$ , and then we have new sequence $b=(a_1, a_1+a_2,a_2,a_2+a_3,a_3, \ldots, a_{n-1}, a_{n-1}+a_n, a_n)$ . Step 2, we do the same thing with sequence b to have the new sequence c again…. And so on. If we do 2013 steps, count the number of the number 2013 appear on the line if a) $n=2$ , $a_1=1, a_2=1000$ b) $n=1000$ , $a_i=i, i=1,2\ldots, 1000$ Sorry for my bad English Moderator says: alternate phrasing here: https://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=516134