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Greece Contests
Greece National Olympiad
2017 Greece National Olympiad
4
4
Part of
2017 Greece National Olympiad
Problems
(1)
Coefficients of Polynomial
Source: 2017 Greece National Olympiad Problem 4
5/2/2017
Let
u
u
u
be the positive root of the equation
x
2
+
x
−
4
=
0
x^2+x-4=0
x
2
+
x
−
4
=
0
. The polynomial
P
(
x
)
=
a
n
x
n
+
a
n
−
1
x
n
−
1
+
.
.
.
+
a
0
P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0
P
(
x
)
=
a
n
x
n
+
a
n
−
1
x
n
−
1
+
...
+
a
0
where
n
n
n
is positive integer has non-negative integer coefficients and
P
(
u
)
=
2017
P(u)=2017
P
(
u
)
=
2017
. 1) Prove that
a
0
+
a
1
+
.
.
.
+
a
n
≡
1
m
o
d
2
a_0+a_1+...+a_n\equiv 1\mod 2
a
0
+
a
1
+
...
+
a
n
≡
1
mod
2
. 2) Find the minimum possible value of
a
0
+
a
1
+
.
.
.
+
a
n
a_0+a_1+...+a_n
a
0
+
a
1
+
...
+
a
n
.
polynomial
algebra