1963 IMO Problems/Problem 5
Prove that .
Because the sum of the -coordinates of the seventh roots of unity is , we have
Now, we can apply to obtain
Finally, since ,
Let . We have
Then, by product-sum formulae, we have
Let and . From the addition formulae, we have
From the Trigonometric Identity, , so
We must prove that . It suffices to show that .
Now note that . We can find these in terms of and :
Therefore . Note that this can be factored:
Clearly , so . This proves the result.
Let . Thus it suffices to show that . Now using the fact that and , this is equivalent to But since is a th root of unity, . The answer is then , as desired.
We let . We therefore have , where , are the roots of unity. Since , then , so . Therefore, because , so
Since , we have and we are done
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