6. On the side $A C$ of triangle $A B C$, there is a point $D$ such that $A D: A C=2: 5$, and $B D+2 B C=3 A B$. The inscribed circle of the triangle with center at point $O$ intersects $B D$ at points $M$ and $N$. Find the angle $M O N$.
Answer: $2 \arccos \frac{1}{5}$.

Examples of tasks from the database of the remote qualifying round of the "Rosatom" Olympiad, 11th grade

The database of the remote qualifying round of the "Rosatom" Olympiad (which is only for 11th-grade students) contains more than 300 problems with numerical answers (which are the ones being checked). These problems are updated annually, new ones are added, and the numbers in each problem are changed. Each participant in the round receives 6 problems randomly. To avoid errors related to rounding the answer (if the answer is not an integer), each such problem specifies a small interval of values, all answers within which are considered correct. To advance to the final round, one needed to solve five out of six problems.