Problem 1611 (difficulty: 9/10)

The sequence \(\displaystyle a_0,a_1,\ldots,\) is defined recursively by \(\displaystyle a_0=-1\) and the requirement \(\displaystyle \displaystyle\sum\limits_{k=0}^n\frac{a_k}{n-k+1}=0\) for all \(\displaystyle n\ge 1\). Show that for all \(\displaystyle n\ge1\) \(\displaystyle a_n>0\). (IMO Shortlist, 2006)

Use complex analysis to solve this probem by showing that

\(\displaystyle a_n = \int_1^\infty\frac{\dx}{x^n(\pi^2+\log^2(x-1))}. \)

Give me another random problem!