5. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ and two planes $\alpha$ and $\beta$. Plane $\alpha$ is perpendicular to the line $A_{1} C_{1}$, and plane $\beta$ is parallel to the line $C D_{1}$. Determine the smallest possible angle between the planes $\alpha$ and $\beta$.

( 7 points)
Answer: $\frac{\pi}{6}$.

Solution: First, let's prove one auxiliary statement. Let planes $\alpha$ and $\beta$ intersect along a line $l$ and form a dihedral angle $\varphi$. OPEN REGIONAL INTERUNIVERSITY OLYMPIAD 2018-2019  MATHEMATICS (11th GRADE)  INCLUSIVE STAGE  VARIANT 2  (ANSWERS)