$\text{ If }\left[x^2-2x+a\right]=0,\text{ where }[ ·]\text{ represents greatest integer function,has no solution then, }$

$\begin{array}{l}\left[x^2-2x+a\right]=0\\ \Rightarrow x^2-2x+a\in [0,1)\end{array}$

For no solution

$\begin{array}{l}\text{ case (i): }\\ x^2-2x+a<0,\mathrm{\forall }x\in R\end{array}$

which is not possible ,since $x^2$ coefficient is 1 is positive and expression is negative.

$\text{ case (ii): }$

$\begin{array}{l}{x}^2-2x+a\ge 1,\mathrm{\forall }x\in R\\ \Rightarrow x^2-2x+a-1\ge 0,\mathrm{\forall }x\in R\\ \Rightarrow \text{ Disc. }\le 0\\ \Rightarrow 4-4(a-1)\le 0\Rightarrow 2\le a\end{array}$