BALKAN 2015 Problem 4

Prove that among any 20 consecutive positive integers there exists an integer $d$ such that for each positive integer $n$ we have the inequality

$$n \sqrt{d}\{n \sqrt{d}\}>\frac{5}{2}$$

where $\{x\}$ denotes the fractional part of the real number $x$. The fractional part of a real number $x$ is $x$ minus the greatest integer less than or equal to $x$.