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Putnam
1947 Putnam
A2
Putnam 1947 A2
Putnam 1947 A2
Source: Putnam 1947
April 3, 2022
Putnam
function
algebra
functional equation
Problem Statement
A real valued continuous function
f
f
f
satisfies for all real
x
x
x
and
y
y
y
the functional equation
f
(
x
2
+
y
2
)
=
f
(
x
)
f
(
y
)
.
f(\sqrt{x^2 +y^2 })= f(x)f(y).
f
(
x
2
+
y
2
)
=
f
(
x
)
f
(
y
)
.
Prove that
f
(
x
)
=
f
(
1
)
x
2
.
f(x) =f(1)^{x^{2}}.
f
(
x
)
=
f
(
1
)
x
2
.
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