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Taiwan Contests
Taiwan National Olympiad
1995 Taiwan National Olympiad
4
4
Part of
1995 Taiwan National Olympiad
Problems
(1)
exists plynomial with two conditions
Source: 4-th Taiwanese Mathematical Olympiad 1995
1/16/2007
Let
m
1
,
m
2
,
.
.
.
,
m
n
m_{1},m_{2},...,m_{n}
m
1
,
m
2
,
...
,
m
n
be mutually distinct integers. Prove that there exists a
f
(
x
)
∈
Z
[
x
]
f(x)\in\mathbb{Z}[x]
f
(
x
)
∈
Z
[
x
]
of degree
n
n
n
satisfying the following two conditions: a)
f
(
m
i
)
=
−
1
∀
i
=
1
,
2
,
.
.
.
,
n
f(m_{i})=-1\forall i=1,2,...,n
f
(
m
i
)
=
−
1∀
i
=
1
,
2
,
...
,
n
. b)
f
(
x
)
f(x)
f
(
x
)
is irreducible.
algebra
polynomial
algebra unsolved