MathDB

A soccer coach named

C

does a header drill with two players

A

and

B

, but they all forgot to put sunscreen on their foreheads. They solve this issue by dunking the ball into a vat of sunscreen before starting the drill. Coach

C

heads the ball to

A

, who heads the ball back to

C

, who then heads the ball to

B

, who heads the ball back to

C

; this pattern

CACBCACB\ldots\,

repeats ad infinitum. Each time a person heads the ball,

1/10

of the sunscreen left on the ball ends up on the person's forehead. In the limit, what fraction of the sunscreen originally on the ball will end up on the coach's forehead?

Problem Statement