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Greece Contests
Greece National Olympiad
2016 Greece National Olympiad
2
2
Part of
2016 Greece National Olympiad
Problems
(1)
Polynomials
Source: Greek mathematical olympiad,Problem 2
2/29/2016
Find all monic polynomials
P
,
Q
P,Q
P
,
Q
which are non-constant, have real coefficients and they satisfy
2
P
(
x
)
=
Q
(
(
x
+
1
)
2
2
)
−
Q
(
(
x
−
1
)
2
2
)
2P(x)=Q(\frac{(x+1)^2}{2})-Q(\frac{(x-1)^2}{2})
2
P
(
x
)
=
Q
(
2
(
x
+
1
)
2
)
−
Q
(
2
(
x
−
1
)
2
)
and
P
(
1
)
=
1
P(1)=1
P
(
1
)
=
1
for all real
x
x
x
.
algebra
polynomial