A sequence $a_1,a_2,\ldots$ of real numbers has the following properties: (i) $|a_1+a_2+\ldots +a_k|\leq 1$ for every $k$; (ii) $|a_k-a_{k-1}|\leq 1/k$ for every $k\geq 2$. Suppose that $\displaystyle |a_k|\geq \frac{c}{\sqrt{k}}$ for infinitely many $k$. Prove that $c\leq \sqrt{2}$. We received a solution from Josiah Moltz and Dr Mathew Wolak. For detailed solutions see the following link {{:pow:2025sproblem4.pdf|Solution}}.