The USSR Olympiad Problem Book
#198. Prove that in the product \begin{align*} \left(1 - x + x^2 - x^3 + \cdots - x^{99} + x^{100}\right)\left(1 + x + x^2 + x^3 + \cdots + x^{99} + x^{100}\right), \end{align*} after multiplying and collecting terms, there does not appear a term in \(x\) of odd degree.
Hint. Multiply out for low \(n\) to see the pattern, prove by induction.