Show that

(a)

For any non-negative integer $k,$ observe that $(10k+1.3)^2=100k^2+26k+1.69.$

Thus, if $x=10k+1.3\quad(k\in\mathbb{Z},\ k\geqslant0),$ we have $\{x\}+\{x^2\}=0.3+0.69=0.99.$

There are infinitely many non-negative integers $k$ and hence infinitely many such rationals $x.$

(b)

Note that for any $x\in\mathbb{R}^+,\ \{x\}=0\ \Leftrightarrow\ x\in\mathbb{Z}.$ Hence, in order to have $\{x\}+\{x^2\}=1,$ we must have $x\notin\mathbb{Z}.$ If $x$ is a positive non-integer rational, $x^2$ would have more decimal places than $x.$ This would mean that their fractional parts cannot add up to 1. Hence there are no positive rationals $x$ such that $\{x\}+\{x^2\}=1,$

Remark: There do however exist positive irrational numbers $x$ with $\{x\}+\{x^2\}=1,$ To see this, note that $f(x)=\{x\}+\{x^2\}$ is continuous on the interval $[0.1\,,\,0.9]$ and $f(0.1)<1$ while $f(0.9)>1;$ the result follows by the intermediate-value theorem.