Question 78
Prove that any set of 20 consecutive positive integers contains an integer $k$ such that $n\sqrt{k} \cdot \{n\sqrt{k}\} \gt 5/2$ for every positive integer $n$. Here $\{x\}$ denotes the fractional part of the real number $x$, that is, the difference of $x$ and the largest integer not exceeding $x$.