Question 208: In 1, 2, ..., 2011, how many numbers can be expressed in the form $[\mathrm{x}[\mathrm{x}]]$, where $[\mathrm{x}]$ denotes the greatest integer not exceeding x.

In 1, 2, ..., 2011, how many numbers can be expressed in the form $[\mathrm{x}[\mathrm{x}]]$, here $[\mathrm{x}]$ represents the greatest integer not exceeding x.
Question 208, Answer: Let $x=m+\alpha$, where $m \in N, \alpha \in[0,1)$, then $[x[x]]=[m(m+\alpha)]=$ $m^{2}+[m \alpha] \in\left\{m^{2}, m^{2}+1, \ldots, m^{2}+m-1\right\}$. Therefore, when $[x]=m$, $[x[x]]$ can take exactly $m$ different values. There are exactly $1+2+\ldots+44=990$ numbers.