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Putnam
1991 Putnam
B2
B2
Part of
1991 Putnam
Problems
(1)
Cute functional equations
Source: 52nd Putnam 1991 Problem B2
11/17/2020
Define functions
f
f
f
and
g
g
g
as nonconstant, differentiable, real-valued functions on
R
R
R
. If
f
(
x
+
y
)
=
f
(
x
)
f
(
y
)
−
g
(
x
)
g
(
y
)
f(x+y)=f(x)f(y)-g(x)g(y)
f
(
x
+
y
)
=
f
(
x
)
f
(
y
)
−
g
(
x
)
g
(
y
)
,
g
(
x
+
y
)
=
f
(
x
)
g
(
y
)
+
g
(
x
)
f
(
y
)
g(x+y)=f(x)g(y)+g(x)f(y)
g
(
x
+
y
)
=
f
(
x
)
g
(
y
)
+
g
(
x
)
f
(
y
)
, and
f
′
(
0
)
=
0
f'(0)=0
f
′
(
0
)
=
0
, prove that
(
f
(
x
)
)
2
+
(
g
(
x
)
)
2
=
1
\left(f(x)\right)^2+\left(g(x)\right)^2=1
(
f
(
x
)
)
2
+
(
g
(
x
)
)
2
=
1
for all
x
x
x
.
function
functional equation
algebra
superior algebra