1963 IMO Problems/Problem 5
Problem
Prove that .
Solution
Because the sum of the -coordinates of the seventh roots of unity is , we have
Now, we can apply to obtain
Finally, since ,
~mathboy100
Solution 2
Let . We have
Then, by product-sum formulae, we have
Thus .
Solution 3
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.
Solution 4
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.
~yofro
Solution 5
We let . We therefore have , where , are the roots of unity. Since , then , so . Therefore, because , so
Since , we have and we are done
See Also
1963 IMO (Problems) • Resources | ||
Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
All IMO Problems and Solutions |