Find the area of the $MNRK$ trapezoid with the lateral side $RK = 3$ if the distances from the vertices $M$ and $N$ to the line $RK$ are $5$ and $7$, respectively.

It turned out for some triangle with sides $a, b$ and $c$, that a circle of radius $r = \frac{a+b+c}{2}$ touches side $c$ and extensions of sides $a$ and $b$. Prove that a circle of radius $ \frac{a+c-b}{2}$ is tangent to $a$ and the extensions of $b$ and $c$.

Jack has a quadrilateral that consists of four sticks. It turned out that Jack can form three different triangles from those sticks. Prove that he can form a fourth triangle that is different from the others.

A regular tetrahedron $SABC$ of volume $V$ is given. The midpoints $D$ and $E$ are taken on $SA$ and $SB$ respectively and the point $F$ is taken on the edge $SC$ such that $SF: FC = 1: 3$. Find the volume of the pentahedron $FDEABC$.