2014 AIME I Problems/Problem 5

Problem 5

Let the set $S = \{P_1, P_2, \dots, P_{12}\}$ 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.)

Solution

By looking at the problem and drawing a few pictures, it quickly becomes obvious that one cannot draw a circle that covers 2 disjoint areas of the 12-gon without including all the vertices in between those areas. In other words, in order for a subset to be communal, all the vertices in the subset must be adjacent to one another. We now count the number of ways to select a row of adjacent vertices. We notice that for any subset size between 1 and 11, there are 12 possible subsets like this (this is true because we can pick any of the 12 vertices as a "starting" vertex, include some number of vertices counterclockwise from that vertex, and generate all possible configurations). However, we also have to include the set of all 12 vertices, as well as the empty set. Thus, the total number is $12*11 + 2 = \boxed{134}$.

See also

2014 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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