Question

Let the set $S = \{P_1, P_2, \ldots, P_{12}\}$ consist of the twelve vertices of a regular $12$ -gon. A subset $Q$ of $S$ is called communal if there is a circle such that all points of $Q$ are inside the circle, and all points of $S$ not in $Q$ are outside of the circle. How many communal subsets are there? (Note that the empty set is a communal subset.)

Answer

Solution:

A subset $Q$ is communal exactly when its vertices are consecutive around the $12$ -gon. Indeed, a separating circle meets the circumcircle of the $12$ -gon in at most two points, so the vertices inside it form a contiguous arc. Conversely, any run of consecutive vertices can be separated from the remaining vertices by a line, and a sufficiently large circle on the proper side of that line contains exactly that run.

For each size $k$ with $1 \le k \le 11$ there are $12$ runs of $k$ consecutive vertices (one starting at each vertex), giving $12 \cdot 11 = 132$ subsets, and the empty set and all of $S$ are also communal. The total is $132 + 2 = 134.$

2014 AIME I Problem 5

Solution:

Problem 5 in Other Years