Task 12. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with side $3 \sqrt{2}$. Find the volume of a regular tetrahedron, one vertex of which coincides with point $A$, and the other three vertices belong to the plane $C M A_{1} N$, where $M$ and $N$ are the midpoints of edges $D D_{1}$ and $B B_{1}$.
Answer: 9

Problem 13. Solve the equations for all natural $n$:

$$
\cos ^{4} x+\sin x(\sin x+1)\left(\cos ^{2} x+\sin x-1\right)=n
$$

In the answer, write the number of roots in the interval $[0,2 \pi]$

Answer: 3

Problem 14. $M$ is the midpoint of the lateral side $A B$ of trapezoid $A B C D$, and $E$ is the intersection point of segments $M D$ and $A C$. Find the area of triangle $A D E$, given that $B C: A D=1: 3$, and the area of trapezoid $A B C D$ is 196.

Answer: 63

Problem 15. Solve the equation

$$
x+P(x)+P(P(x))+P(P(P(x)))=2014
$$

where $P(x)$ is the sum of the digits of the natural number $x$. In the answer, write the sum of the roots of this equation.

Answer: 3986