Problem 10.4. Find all values of the real parameter $a$ such that the number of the solutions of the equation

$$
3\left(5 x^{2}-a^{4}\right)-2 x=2 a^{2}(6 x-1)
$$

does not exceed the number of the solutions of the equation

$$
2 x^{3}+6 x=\left(3^{6 a}-9\right) \sqrt{2^{8 a}-\frac{1}{6}}-(3 a-1)^{2} 12^{x}
$$

Ivan Landjev
10.4. The first equation is quadratic with discriminant $D=\left(9 a^{2}-1\right)^{2}$. Therefore it has two different solutions for $a \neq \pm \frac{1}{3}$ and exactly one solution for $a= \pm \frac{1}{3}$.

Since the function

$$
2 x^{3}+6 x+(3 a-1)^{2} 12^{x}
$$

is strictly increasing, the second equation has at most one solution. For $a=\frac{1}{3}$ it has solution $x=0$, while for $a=-\frac{1}{3}$ it is not defined. Finally, the only value of $a$ which satisfies the condition, is $a=\frac{1}{3}$.