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Problems
Contests
National and Regional Contests
Iran Contests
Simurgh
2019 Simurgh
4
Polynomial
Polynomial
Source: Simurgh 2019 - Problem 4
February 24, 2019
algebra
polynomial
Problem Statement
Assume that every root of polynomial
P
(
x
)
=
x
d
−
a
1
x
d
−
1
+
.
.
.
+
(
−
1
)
d
−
k
a
d
P(x) = x^d - a_1x^{d-1} + ... + (-1)^{d-k}a_d
P
(
x
)
=
x
d
−
a
1
x
d
−
1
+
...
+
(
−
1
)
d
−
k
a
d
is in
[
0
,
1
]
[0,1]
[
0
,
1
]
. Show that for every
k
=
1
,
2
,
.
.
.
,
d
k = 1,2,...,d
k
=
1
,
2
,
...
,
d
the following inequality holds:
a
k
−
a
k
+
1
+
.
.
.
+
(
−
1
)
d
−
k
a
d
≥
0
a_k - a_{k+1} + ... + (-1)^{d-k}a_d \geq 0
a
k
−
a
k
+
1
+
...
+
(
−
1
)
d
−
k
a
d
≥
0
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