Problem 1612
Let \(\displaystyle \displaystyle f(z)=\sum_{n=0}^\infty a_nz^n\) be entire that satisfies \(\displaystyle |f(z)|<e^{|z|}\). Prove that \(\displaystyle |a_n|\le\left(\frac{e}{n}\right)^n\). Difficulty: 5. |
Problem 1611
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 \(\displaystyle a_n = \int_1^\infty\frac{\dx}{x^n(\pi^2+\log^2(x-1))}. \) Difficulty: 9. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |