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All-Russian Olympiad Regional Round
2004 All-Russian Olympiad Regional Round
11.3
11.3
Part of
2004 All-Russian Olympiad Regional Round
Problems
(1)
P(x) = a_nx^n+a_{n-1}x^{n-1}+...+a_0 - All-Russian MO 2004 Regional (R4) 11.3
Source:
9/27/2024
Let 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
has at least one real root and
a
0
≠
0
a_0 \ne 0
a
0
=
0
. Prove that, consequently crossing out the monomials in the notation
P
(
x
)
P(x)
P
(
x
)
in some order, we can obtain the number
a
0
a_0
a
0
from it so that each intermediate polynomial also has at least one real root.
algebra
polynomial