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IMO Shortlist 1969 problem 6

1969 shortlist
(BEL 6) Evaluate \left(\cos\frac{\pi}{4} + i \sin\frac{\pi}{4}\right)^{10} in two different ways and prove that \dbinom{10}{1}-\dbinom{10}{3}+\frac{1}{2}\dbinom{10}{5}=2^4

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