# Problem 6.

B-1 For which values of the parameter $a \in \mathbb{R}$ does the maximum distance between the roots of the equation

$$
a \operatorname{tg}^{3} x+\left(2-a-a^{2}\right) \operatorname{tg}^{2} x+\left(a^{2}-2 a-2\right) \operatorname{tg} x+2 a=0
$$

belonging to the interval $\left(-\frac{\pi}{2} ; \frac{\pi}{2}\right)$, take the smallest value? Find this smallest value.
Answer: $\frac{\pi}{4}$ when $a=0$.

Solution. The given equation can be rewritten as $(\operatorname{tg} x-1)(\operatorname{tg} x-a)(a \operatorname{tg} x+2)=0$, from which, for $x \in\left(-\frac{\pi}{2} ; \frac{\pi}{2}\right)$, either $\operatorname{tg} x=1$ and $x=\frac{\pi}{4}$, or $\operatorname{tg} x=a$ and $x=\operatorname{arctg} a$, or (for $a \neq 0$) $\operatorname{tg} x=-\frac{2}{a}$ and $x=-\operatorname{arctg} \frac{2}{a}$. Thus, the given equation has two or three different roots on the interval $\left(-\frac{\pi}{2} ; \frac{\pi}{2}\right)$ (the second root cannot coincide with the third, as $\operatorname{arctg} a$ and $-\operatorname{arctg} \frac{2}{a}$ have different signs for any $a \neq 0$ due to the oddness of the arctangent).

Case 1: $a=0$. Then there are two roots $x=\frac{\pi}{4}$ and $x=0$, which differ by $\frac{\pi}{4}$.

Case 2: $a>0$. Then the difference between the roots $x=\frac{\pi}{4}$ and $x=-\operatorname{arctg} \frac{2}{a}<0$ is greater than $\frac{\pi}{4}$.

Case 3: $a<0$. Then the difference between the roots $x=\frac{\pi}{4}$ and $x=\operatorname{arctg} a<0$ is greater than $\frac{\pi}{4}$.

B-2 For which values of the parameter $a \in \mathbb{R}$ does the greatest distance between the roots (not necessarily adjacent) of the equation

$$
a \operatorname{tg}^{3} x+\left(1-a-2 a^{2}\right) \operatorname{tg}^{2} x+\left(2 a^{2}-2 a-1\right) \operatorname{tg} x+2 a=0,
$$

belonging to the interval $\left(-\frac{\pi}{2} ; \frac{\pi}{2}\right)$, take the smallest value? Find this smallest value.

Answer: $\frac{\pi}{4}$ when $a=0$.

B-3 For which values of the parameter $a \in \mathbb{R}$ does the greatest distance between the roots (not necessarily adjacent) of the equation

$$
a \operatorname{ctg}^{3} x+\left(2 a^{2}-a-2\right) \operatorname{ctg}^{2} x+\left(2-4 a-2 a^{2}\right) \operatorname{ctg} x+4 a=0
$$

belonging to the interval $(0 ; \pi)$, take the smallest value? Find this smallest value.

Answer: $\frac{\pi}{4}$ when $a=0$.

B-4 For which values of the parameter $a \in \mathbb{R}$ does the greatest distance between the roots (not necessarily adjacent) of the equation

$$
a \operatorname{ctg}^{3} x+\left(a^{2}-a-3\right) \operatorname{ctg}^{2} x+\left(3-3 a-a^{2}\right) \operatorname{ctg} x+3 a=0
$$

belonging to the interval $(0 ; \pi)$, take the smallest value? Find this smallest value.

Answer: $\frac{\pi}{4}$ when $a=0$.

## Lomonosov Olympiad for Schoolchildren in Mathematics

Final stage $2021 / 22$ academic year for 11th grade

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