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Putnam
1981 Putnam
A4
A4
Part of
1981 Putnam
Problems
(1)
Putnam 1981 A4
Source: Putnam 1981
3/31/2022
A point
P
P
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
)
N( t)
N
(
t
)
be the number of starting directions from a fixed interior point
P
0
P_0
P
0
for which
P
P
P
escapes within
t
t
t
units of time. Find the least constant
a
a
a
for which constants
b
b
b
and
c
c
c
exist such that
N
(
t
)
≤
a
t
2
+
b
t
+
c
N(t) \leq at^2 +bt+c
N
(
t
)
≤
a
t
2
+
b
t
+
c
for all
t
>
0
t>0
t
>
0
and all initial points
P
0
.
P_0 .
P
0
.
Putnam
reflection
geometry