Question 132
a) Prove that for any rational number $\alpha \in (0; 1)$ there exists an infinite set of real numbers that satisfy the equation $\{x[x\{x\}]\} = \alpha$ and any two of them have the same fractional part.
b) Prove that for any rational number $\alpha \in (0; 1)$ there exists an infinite set of real numbers that satisfy the equation $\{x[x\{x\}]\} = \alpha$ and any two of them have different fractional parts.
(The fractional part of a real number $a$ is given by $\{a\} = a - \lfloor a \rfloor$, where $\lfloor a \rfloor$ is its integer part, i.e., the greatest integer that does not exceed $a$.)
b) Prove that for any rational number $\alpha \in (0; 1)$ there exists an infinite set of real numbers that satisfy the equation $\{x[x\{x\}]\} = \alpha$ and any two of them have different fractional parts.
(The fractional part of a real number $a$ is given by $\{a\} = a - \lfloor a \rfloor$, where $\lfloor a \rfloor$ is its integer part, i.e., the greatest integer that does not exceed $a$.)