6. Point $M$ lies on the edge $A B$ of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$. A rectangle $M N L K$ is inscribed in the square $A B C D$ such that one of its vertices is point $M$, and the other three are located on different sides of the square base. The rectangle $M_{1} N_{1} L_{1} K_{1}$ is the orthogonal projection of the rectangle $M N L K$ onto the plane of the upper base $A_{1} B_{1} C_{1} D_{1}$. The plane of the quadrilateral $M K_{1} L_{1} N$ forms an angle $\alpha$ with the plane of the base of the cube, where $\cos \alpha=\sqrt{\frac{2}{11}}$. Find the ratio $A M: M B$
Answer: $A M: M B=1: 2$

## Evaluation Criteria Final Rosatom 11th grade