11.4. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge length a. The ends of the segment intersecting edge $C_{1} D_{1}$ lie on the lines $A A_{1}$ and $B C$. What is the minimum length that this segment can have
11.4. Let points $M$ and $N$ lie on the lines $A A_{1}$ and $B C$ respectively, and the segment $M N$ intersects the line $C_{1} D_{1}$ at point $L$. Then points $M$ and $N$ lie on the rays $A A_{1}$ and $B C$, with $x = A M > a$ and $y = B N > a$. Considering projections onto the planes $A A_{1} B$ and $A B C$, we get $C_{1} L: L D_{1} = a: (x - a)$ and $C_{1} L: L D_{1} = (y - a): a$. Therefore, $(x - a)(y - a) = a^{2}$, i.e., $x y = (x + y) a$, and thus, $(x y)^{2} = (x + y)^{2} a^{2} \geq 4 x y a^{2}$, i.e., $x y \geq 4 a^{2}$. Consequently, $M N^{2} = x^{2} + y^{2} + a^{2} = (x + y)^{2} - 2 x y + a^{2} = (x y)^{2} / a^{2} - 2 x y + a^{2} = (x y - a^{2})^{2} / a^{2} \geq 9 a^{2}$. The minimum value of the length of segment $M N$ is $3 a$; it is achieved when $A M = B N = 2 a$.