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1979 Polish MO Finals
6
sum 1 / w′(x_i)=0 for polynomial with x_i dinstinct roots
sum 1 / w′(x_i)=0 for polynomial with x_i dinstinct roots
Source: Polish MO Finals 1979 p6
August 24, 2024
algebra
polynomial
Problem Statement
A polynomial
w
w
w
of degree
n
>
1
n > 1
n
>
1
has
n
n
n
distinct zeros
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
. Prove that:
1
w
′
(
x
1
)
+
1
w
′
(
x
2
)
+
.
.
.
⋅
⋅
⋅
+
1
w
′
(
x
n
)
=
0.
\frac{1}{w'(x_1)}+\frac{1}{w'(x_2)}+...···+\frac{1}{w'(x_n)}= 0.
w
′
(
x
1
)
1
+
w
′
(
x
2
)
1
+
...
⋅⋅⋅
+
w
′
(
x
n
)
1
=
0.
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