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National and Regional Contests
Moldova Contests
Moldova National Olympiad
2006 Moldova National Olympiad
12.4
12.4
Part of
2006 Moldova National Olympiad
Problems
(1)
P(-1) is real
Source: Moldavian Republic Olympiad
3/4/2006
Let
P
(
x
)
=
x
n
+
a
1
x
n
−
1
+
.
.
.
+
a
n
−
1
x
+
(
−
1
)
n
P(x)= x^n+a_{1}x^{n-1}+...+a_{n-1}x+(-1)^{n}
P
(
x
)
=
x
n
+
a
1
x
n
−
1
+
...
+
a
n
−
1
x
+
(
−
1
)
n
,
a
i
∈
C
a_{i} \in C
a
i
∈
C
,
n
≥
2
n\geq 2
n
≥
2
with all roots having same modulo. Prove that
P
(
−
1
)
∈
R
P(-1) \in R
P
(
−
1
)
∈
R
algebra proposed
algebra