Problem: (2007 Mock AIME 6 – #7) Let and for all integers . How many more distinct complex roots does have than ?
Solution: Well, is familiar. Rewrite as
i.e. the th roots of unity except . Since just contains a term that doesn’t have, we only need to think about that term in relation to the previous ones. So we want to find out how many of the th roots of unity are not -th roots of unity for any . Well, recalling that any -th root of unity can be written as
for , it remains to find the number of such that does not reduce. But this, of course, is simply . QED.
Comment: Definitely one of my favorite problems on the test because it has a really nice and clean solution once you see it. And it wasn’t too difficult to see either; combining a little knowledge of roots of unity with a little knowledge of the function made it a good problem. Furthermore, it offers the nice generalization that
has more distinct roots than .
Practice Problem: (2007 Mock AIME 6 – #8) A sequence of positive reals is defined by , , and for all integers . Given that and , find .
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