5. After substituting the variables $X=\sqrt{|x-1|}, \sqrt{3|y|}$, we obtain the system:

$$
\left\{\begin{array}{l}
X+Y=1 \\
X^{4}+Y^{4}=-a \\
X \geqslant 0, Y \geqslant 0
\end{array}\right.
$$

Each solution $\left(X_{0}, Y_{0}\right)$ of this system such that $0<X_{0}<1,0<Y_{0}<1, X_{0} \neq Y_{0}$ generates exactly four distinct solutions of the original system. Therefore, either $X_{0}=$ $Y_{0}=\frac{1}{2}$, in which case $-a=\frac{1}{2^{4}}+\frac{1}{2^{4}}=\frac{1}{8}$, or $X_{0}=1, Y_{0}=0$ and $X_{0}=0, Y_{0}=1$, in which case $-a=1+0=0+1$.

Verification. Consider the function $f(t)=t^{4}+(1-t)^{4}$. Since the derivative $f^{\prime}(t)=4\left(t^{3}-(1-t)^{3}\right)$ is zero only at the point $t=1 / 2$, for $t \in[0 ; 1]$ we obtain the estimate $\frac{1}{8}=f\left(\frac{1}{2}\right) \leqslant f(t) \leqslant f(1)=f(0)=1$. Therefore, when $a=-\frac{1}{8}$, there is only one solution of the system satisfying $(X, Y), X \geqslant 0, Y \geqslant 0$, namely $\left(X_{0}, Y_{0}\right)=\left(\frac{1}{2}, \frac{1}{2}\right)$, which generates four solutions of the original system. When $a=-1$, there are only two solutions of the system satisfying $(X, Y), X \geqslant 0, Y \geqslant 0$, namely $\left(X_{0}, Y_{0}\right)=(0,1),(1,0)$, which also generates four solutions of the original system.
Answer: $a=-1, a=-1 / 8$. Answer to the option: $4-2: a=1 / 2, a=1 / 16$.

Answer and solution to option $5-1$ (2) 1. Answer: $x= \pm \frac{\pi}{12}+\frac{\pi n}{2}$. Since $\operatorname{ctg}^{2} x-\operatorname{tg}^{2} x=\frac{4 \cos 2 x}{\sin ^{2} 2 x}$, the original equation is equivalent to the equation

$$
\frac{4 \cos 2 x}{\sin ^{2} 2 x}=\frac{12}{\cos 2 x} \Leftrightarrow\left\{\begin{array}{l}
\cos ^{2} 2 x=3 \sin ^{2} 2 x \\
\sin x \neq 0, \cos x \neq 0, \cos 2 x \neq 0
\end{array}\right.
$$

Solving the equation of the system, we get $\operatorname{tg}^{2} 2 x=\frac{1}{3} \Leftrightarrow \operatorname{tg} 2 x= \pm \frac{1}{\sqrt{3}}$, from which $x= \pm \frac{\pi}{12}+\frac{\pi n}{2}$. All found values of $x$ satisfy the other conditions of the system.