Let a1, a2, . . . , ar be r not necessarily distinct positive integers such that
(x + a1)(x + a2) · · · (x + ar) = xr + 230xr−1 + · · · + 2007,
where each missing term on the right has degree between 1 and r − 2
inclusive. What is the value of r ?
Answers
A) 4
B) 12
C) 63
D) 669
E) 2007
The conditions imply a1 + a2 + · · · + ar = 230 and a1a2 · · · ar = 2007. One checks that   2007 = 32 · 223, where 223 is prime. We deduce that some aj is 223, two aj’s are 3, and the   remaining aj’s are 1. Since their sum is 230 = 223 + 3 + 3 + 1, we obtain r = 4.   correct answer 4
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Let a1, a2, . . . , ar be r not necessarily distinct positive integers such that
(x + a1)(x + a2) · · · (x + ar) = xr + 230xr−1 + · · · + 2007,
where each missing term on the right has degree between 1 and r − 2
inclusive. What is the value of r ?