RO-TST 2015 Problem 12

If $k$ and $n$ are positive integers , and $k \leq n$ , let $M(n,k)$ denote the least common multiple of the numbers $n , n-1 , \ldots , n-k+1$ .Let $f(n)$ be the largest positive integer $k \leq n$ such that $M(n,1)<M(n,2)<\ldots <M(n,k)$ . Prove that :(a) $f(n)<3\sqrt{n}$ for all positive integers $n$ .(b) If $N$ is a positive integer , then $f(n) > N$ for all but finitely many positive integers $n$ .