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Miklós Schweitzer
1997 Miklós Schweitzer
3
polynomial
polynomial
Source: miklos schweitzer 1997 q3
September 23, 2021
polynomial
algebra
Miklos Schweitzer
Problem Statement
Denote
f
n
(
X
)
∈
Z
[
X
]
f_n(X) \in \Bbb Z [X]
f
n
(
X
)
∈
Z
[
X
]
the polynomial
Π
j
=
1
n
(
X
+
j
−
1
)
\Pi_{j=1}^n ( X + j -1)
Π
j
=
1
n
(
X
+
j
−
1
)
. Show that if the numbers
α
\alpha
α
and
β
\beta
β
satisfy
f
1997
′
(
α
)
=
f
1999
′
(
β
)
=
0
f'_{1997} (\alpha) = f'_{1999} (\beta) = 0
f
1997
′
(
α
)
=
f
1999
′
(
β
)
=
0
, then
f
1997
(
α
)
≠
f
1999
(
β
)
f_{1997} (\alpha ) \neq f_{1999} (\beta)
f
1997
(
α
)
=
f
1999
(
β
)
.
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