1963 IMO Problems/Problem 5
Problem
Prove that .
Solution 1
Let . We have
Then, by product-sum formulae, we have
Thus .
Solution 2
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 3
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
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 |