Question 30
Consider the equations
$$
[x]^3 + x^2 = x^3 + [x]^2 \text{ and } \left[x^3\right] + x^2 = x^3 + \left[x^2\right]
$$
where $[t]$ is the greatest integer that does not exceed $t$. Prove that:
a) any solution of the first equation is an integer;
b) the second equation has a non-integer solution.
$$
[x]^3 + x^2 = x^3 + [x]^2 \text{ and } \left[x^3\right] + x^2 = x^3 + \left[x^2\right]
$$
where $[t]$ is the greatest integer that does not exceed $t$. Prove that:
a) any solution of the first equation is an integer;
b) the second equation has a non-integer solution.