7

Part of 2010 AIME Problems

Problems(2)

Minimally Intersecting

Source: AIME 2010I Problem 7

Define an ordered triple

(A, B, C)

of sets to be minimally intersecting if |A \cap B| \equal{} |B \cap C| \equal{} |C \cap A| \equal{} 1 and A \cap B \cap C \equal{} \emptyset. For example,

(\{1,2\},\{2,3\},\{1,3,4\})

is a minimally intersecting triple. Let

N

be the number of minimally intersecting ordered triples of sets for which each set is a subset of

\{1,2,3,4,5,6,7\}

. Find the remainder when

N

1000

|S|

represents the number of elements in the set

S

Imaginary roots

Source: 2010 AIMEII Problem 7

Let P(z) \equal{} z^3 \plus{} az^2 \plus{} bz \plus{} c, where

a

b

c

are real. There exists a complex number

w

such that the three roots of

P(z)

are w \plus{} 3i, w \plus{} 9i, and 2w \minus{} 4, where i^2 \equal{} \minus{} 1. Find |a \plus{} b \plus{} c|.

algebrapolynomialcomplex numbersAMC