2003 AIME II Problems/Problem 15
[quote=2003 AIME II #15]Let . Let be the distinct zeros of and let for where and are real numbers. Let
where and are integers and is not divisible by the square of any prime. Find [/quote]
We can rewrite the definition of as follows:
This can quite obviously be factored as:
Note that . So the roots of are exactly all -th complex roots of , except for the root .
Let . Then the distinct zeros of are .
We can clearly ignore the root as it does not contribute to the value that we need to compute.
The squares of the other roots are .
Hence we need to compute the following sum:
Using basic properties of the sine function, we can simplify this to
The five-element sum is just . We know that , , and . Hence our sum evaluates to:
Therefore the answer is .
Note that . Our sum can be reformed as
And we can proceed as above.
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