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 $M_k(n)$ 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 $k\geq 1$.
Hint: Take the limit before finding the largest root.