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Baltic Way
2018 Baltic Way
5
5
Part of
2018 Baltic Way
Problems
(1)
Generating polynomials
Source: Baltic Way 2018, Problem 5
11/6/2018
A polynomial
f
(
x
)
f(x)
f
(
x
)
with real coefficients is called generating, if for each polynomial
φ
(
x
)
\varphi(x)
φ
(
x
)
with real coefficients there exists a positive integer
k
k
k
and polynomials
g
1
(
x
)
,
…
,
g
k
(
x
)
g_1(x),\dotsc,g_k(x)
g
1
(
x
)
,
…
,
g
k
(
x
)
with real coefficients such that
φ
(
x
)
=
f
(
g
1
(
x
)
)
+
…
+
f
(
g
k
(
x
)
)
.
\varphi(x)=f(g_1(x))+\dotsc+f(g_k(x)).
φ
(
x
)
=
f
(
g
1
(
x
))
+
…
+
f
(
g
k
(
x
))
.
Find all generating polynomials.
algebra
polynomial