Let denote the value of the integral above. What is the sum of digits of
Define a sequence of functions as follows: Evaluate:
Suppose a light source is positioned in at the origin Smooth mirrors are positioned along the lines and from to The light source is then oriented so that the beam of light emanating from it makes an angle, chosen uniformly and at random, of between and with the positive -axis. The beam is then allowed to reflect back and forth between the two mirrors until it "exits the tunnel", that is, it crosses the line
If the expected distance that the beam of light travels from its source until it exits the tunnel can be expressed as where are positive integers with square-free, then find
A function satisfies the functional equation for all real . If is equal to for positive integers and , then what is the value of ?