14

Part of 2012 AIME Problems

Problems(2)

Complex numbers in geometry

Source: 2012 AIME I Problem 14

Complex numbers

a

b

c

are the zeros of a polynomial

P(z) = z^3+qz+r

|a|^2+|b|^2+|c|^2=250

. The points corresponding to

a

b

c

in the complex plane are the vertices of a right triangle with hypotenuse

h

h^2

geometryalgebrapolynomialVietaanalytic geometrygraphing linesslope

Handshaking in Groups

Source: 2012 AIME II Problem 14

In a group of nine people each person shakes hands with exactly two of the other people from the group. Let N be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the remainder when N is divided by 1000.

HMMTAMCaime 2012