MathDB

5

Part of 2014 AIME Problems

Problems(2)

Communal Subsets

Source: 2014 AIME I Problem 5

3/14/2014
Let the set S={P1,P2,⋯ ,P12}S = \{P_1, P_2, \cdots, P_{12}\} consist of the twelve vertices of a regular 1212-gon. A subset QQ of SS is called communal if there is a circle such that all points of QQ are inside the circle, and all points of SS not in QQ are outside of the circle. How many communal subsets are there? (Note that the empty set is a communal subset.)
geometrycircumcircleAMC
Roots of a Function

Source: 2014 AIME II #5

3/27/2014
Real numbers rr and ss are roots of p(x)=x3+ax+bp(x)=x^3+ax+b, and r+4r+4 and s−3s-3 are roots of q(x)=x3+ax+b+240q(x)=x^3+ax+b+240. Find the sum of all possible values of ∣b∣|b|.
functionalgebrapolynomialquadraticsVietaemailAMC