Mock AIME 3 2006-2007 Problems/Problem 4


Let $x=(i-11)(i-10)....(i+8)(i+9)$ where $i=\sqrt{-1}$. If $a$ and $b$ are positive integers such that $x(a-bi)$ is an integer, then find the minimum value of $a+b$


Pairing the terms $(i-k)$ with $(i+k)$ for $k=1,2,...9$ all result in integers, so we're left with making $i(i-10)(i-11)$ an integer.

$i(i-10)(i-11)=(21+109i)$. To make this an integer, we just multiply it by its conjugate $(21-109i)$, so $a+b=21+109=\boxed{130}$

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