Mock Geometry AIME 2011 Problems/Problem 2

Problem

Eleven nonparallel lines lie on a plane, and their pairwise intersections meet at angles of integer degree. How many possible values are there for the smallest of these angles?

Solution

Consider two intersecting lines. One line can be translated across the plane, without rotation, and the angle between the two lines will remain the same.

Now consider the full eleven lines. These lines can be translated so that they are all concurrent at a point, and still have the same equal pairwise angles.

WLOG, let one of the lines be horizontal. $11$ angles are formed above it by $10$ lines, and all of these angles must sum to $180^\circ$ . Thus, the smallest angle must be less than $\frac{180^\circ}{11}$ , since otherwise they would sum to more than $180^\circ$ . There are $16$ positive integers less than $\frac{180}{11}$ , so there are $\boxed{016}$ possible measures for the smallest angle.

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Mock Geometry AIME 2011 Problems/Problem 2

Problem

Solution