MathDB

A point

P

moves inside a unit square in a straight line at unit speed. When it meets a corner it escapes. When it meets an edge its line of motion is reflected so that the angle of incidence equals the angle of reflection. Let

N( t)

be the number of starting directions from a fixed interior point

P_0

for which

P

escapes within

t

units of time. Find the least constant

a

for which constants

b

and

c

exist such that

N(t) \leq at^2 +bt+c

for all

t>0

and all initial points

P_0 .

Problem Statement