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Putnam 1985 A6

Source:

August 5, 2019

Putnamalgebrapolynomial

Problem Statement

If

p(x)=a_{0}+a_{1} x+\cdots+a_{m} x^{m}

is a polynomial with real coefficients

a_{i},

then set

\Gamma(p(x))=a_{0}^{2}+a_{1}^{2}+\cdots+a_{m}^{2}.

Let

F(x)=3 x^{2}+7 x+2 .

Find, with proof, a polynomial

g(x)

with real coefficients such that
(i)

g(0)=1,

and (ii)

\Gamma\left(f(x)^{n}\right)=\Gamma\left(g(x)^{n}\right)

for every integer

n \geq 1.

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