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Putnam
1985 Putnam
A6
A6
Part of
1985 Putnam
Problems
(1)
Putnam 1985 A6
Source:
8/5/2019
If
p
(
x
)
=
a
0
+
a
1
x
+
⋯
+
a
m
x
m
p(x)=a_{0}+a_{1} x+\cdots+a_{m} x^{m}
p
(
x
)
=
a
0
+
a
1
x
+
⋯
+
a
m
x
m
is a polynomial with real coefficients
a
i
,
a_{i},
a
i
,
then set
Γ
(
p
(
x
)
)
=
a
0
2
+
a
1
2
+
⋯
+
a
m
2
.
\Gamma(p(x))=a_{0}^{2}+a_{1}^{2}+\cdots+a_{m}^{2}.
Γ
(
p
(
x
))
=
a
0
2
+
a
1
2
+
⋯
+
a
m
2
.
Let
F
(
x
)
=
3
x
2
+
7
x
+
2.
F(x)=3 x^{2}+7 x+2 .
F
(
x
)
=
3
x
2
+
7
x
+
2.
Find, with proof, a polynomial
g
(
x
)
g(x)
g
(
x
)
with real coefficients such that(i)
g
(
0
)
=
1
,
g(0)=1,
g
(
0
)
=
1
,
and (ii)
Γ
(
f
(
x
)
n
)
=
Γ
(
g
(
x
)
n
)
\Gamma\left(f(x)^{n}\right)=\Gamma\left(g(x)^{n}\right)
Γ
(
f
(
x
)
n
)
=
Γ
(
g
(
x
)
n
)
for every integer
n
≥
1.
n \geq 1.
n
≥
1.
Putnam
algebra
polynomial