Problem 2. (2 points)

On a circle, 10 points are marked. Any three of them form three inscribed angles. Petya calculated the number of different values that these angles can take. What is the maximum number he could have obtained?
Answer: 80

Solution:

Any two points form two arcs. All inscribed angles subtending the same arc are equal. For two adjacent points, on one of the two arcs between them, no other point lies, meaning a pair of adjacent points gives us one possible angle value, while a pair of non-adjacent points gives two values.

In total, we have 45 pairs of points, of which 35 pairs are non-adjacent. We get the answer $35 \cdot 2 + 10 = 80$.

An example clearly exists. It is sufficient to take the lengths of the arcs between adjacent points in the ratio $1: 2: 4: 8 \ldots$