BALTIC-WAY 2012 Problem 3

a) Show that the equation

$$\lfloor x\rfloor\left(x^{2}+1\right)=x^{3},$$

where $\lfloor x\rfloor$ denotes the largest integer not larger than $x$, has exactly one real solution in each interval between consecutive positive integers.

b) Show that none of the positive real solutions of this equation is rational.