8.5. Kuzya cut a convex paper 67-gon along a straight line into two polygons, then similarly cut one of the two resulting polygons, then one of the three resulting ones, and so on. In the end, he got eight $n$-gons. Find all possible values of $n$.
Answer: $n=11$.

Solution. A straight-line cut can be of three types: from side to side, from a vertex to a side, and from a vertex to a vertex. Therefore, after one cut, the total number of sides of the polygons increases by 4, 3, or 2, respectively. Kuzya made 7 cuts, so the number of sides added is at least 14 but no more than 28. Therefore, the total number of sides of the eight $n$-gons is between 81 and 95. Among the integers in this range, only the number 88 is divisible by 8 without a remainder, so $n=88 \div 8=11$.

Grading Criteria.

“+” A complete and well-reasoned solution is provided

“土” A generally correct reasoning is provided, but it contains minor gaps or inaccuracies (for example, it is not explained how the total number of sides changes in each of the three cases)

“干” The correct answer is obtained, but not all cases are considered

“干” The correct answer is provided and it is only verified that it satisfies the condition, but there is no proof of the absence of other answers

“-” Only the answer is provided

“-” The problem is not solved or is solved incorrectly