Find the smallest possible integer

Square root is an increasing function, so we only need to find an integer $n\ge100$ such that $\{\sqrt{n-1}\} \le 0.5$ but $\{\sqrt{n}\} >0.5$.  Since $\sqrt{110} = 10.488+$ and $\sqrt{111} = 10.535+$ so the smallest such $n$ is 111.