Školjka

%V0 Let $ABCD$ be a regular tetrahedron. To an arbitrary point $M$ on one edge, say $CD$, corresponds the point $P = P(M)$ which is the intersection of two lines $AH$ and $BK$, drawn from $A$ orthogonally to $BM$ and from $B$ orthogonally to $AM$. What is the locus of $P$ when $M$ varies ?

%V0 Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $\frac{1}{8}.$

%V0 Prove that for every point $M$ on the surface of a regular tetrahedron there exists a point $M'$ such that there are at least three different curves on the surface joining $M$ to $M'$ with the smallest possible length among all curves on the surface joining $M$ to $M'$.

%V0 A tetrahedron $ABCD$ is given such that $AD = BC = a; AC = BD = b; AB\cdot CD = c^2$. Let $f(P) = AP + BP + CP + DP$, where $P$ is an arbitrary point in space. Compute the least value of $f(P).$

%V0 Let $ABCD$ be a tetrahedron having each sum of opposite sides equal to $1$. Prove that $$r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}$$ where $r_A, r_B, r_C, r_D$ are the inradii of the faces, equality holding only if $ABCD$ is regular.

%V0 Let $P$ be a point inside a regular tetrahedron $T$ of unit volume. The four planes passing through $P$ and parallel to the faces of $T$ partition $T$ into 14 pieces. Let $f(P)$ be the joint volume of those pieces that are neither a tetrahedron nor a parallelepiped (i.e., pieces adjacent to an edge but not to a vertex). Find the exact bounds for $f(P)$ as $P$ varies over $T.$