BALKAN-SL 2015 N6

Prove that among $20$ consecutive positive integers there is an integer $d$ such that for every positive integer $n$ the following inequality holds

$$n \sqrt{d} \left\{n \sqrt {d} \right \} > \dfrac{5}{2}$$

where by $\left \{x \right \}$ denotes the fractional part of the real number $x$. The fractional part of the real number $x$ is defined as the difference between the largest integer that is less than or equal to $x$ to the actual number $x$.

(Serbia)