Some challenge problems inspired by lectures
Problem: (Inspired by the example from class, from Putnam 2016) Let k be a fixed positive integer. For a positive integer n let Mk(n) be the largest integer m such that \displaystyle \binom{m}{n-k}>\binom{m-k}{n}. Show that \lim_{n\to\infty}\frac{M_k(n)}{n}=\frac{3+\sqrt{5}}{2}, independent of k\geq 1.