MathDB

Robert is a robot who can move freely on the unit circle and its interior, but is attached to the origin by a retractable cord such that at any moment the cord lies in a straight line on the ground connecting Robert to the origin. Whenever his movement is counterclockwise (relative to the origin), the cord leaves a coating of black paint on the ground, and whenever his movement is clockwise, the cord leaves a coating of orange paint on the ground. The paint is dispensed regardless of whether there is already paint on the ground. The paints covers

1

gallon/unit

^2

, and Robert starts at

(1, 0)

. Each second, he moves in a straight line from the point

(\cos(\theta),\sin(\theta))

to the point

(\cos(\theta+a),\sin(\theta+a))

, where a changes after each movement. a starts out as

253^o

and decreases by

2^o

each step. If he takes

89

steps, then the difference, in gallons, between the amount of black paint used and orange paint used can be written as

\frac{\sqrt{a}- \sqrt{b}}{c} \cot 1^o

, where

a, b

and

c

are positive integers and no prime divisor of

c

divides both

a

and

b

twice. Find

a + b + c

Problem Statement