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China Girls Math Olympiad
2021 China Girls Math Olympiad
8
8
Part of
2021 China Girls Math Olympiad
Problems
(1)
CGMO 2021 P8
Source: CGMO 2021 P8
8/14/2021
Let
m
,
n
m, n
m
,
n
be positive integers, define:
f
(
x
)
=
(
x
−
1
)
(
x
2
−
1
)
⋯
(
x
m
−
1
)
f(x)=(x-1)(x^2-1)\cdots(x^m-1)
f
(
x
)
=
(
x
−
1
)
(
x
2
−
1
)
⋯
(
x
m
−
1
)
,
g
(
x
)
=
(
x
n
+
1
−
1
)
(
x
n
+
2
−
1
)
⋯
(
x
n
+
m
−
1
)
g(x)=(x^{n+1}-1)(x^{n+2}-1)\cdots(x^{n+m}-1)
g
(
x
)
=
(
x
n
+
1
−
1
)
(
x
n
+
2
−
1
)
⋯
(
x
n
+
m
−
1
)
.Show that there exists a polynomial
h
(
x
)
h(x)
h
(
x
)
of degree
m
n
mn
mn
such that
f
(
x
)
h
(
x
)
=
g
(
x
)
f(x)h(x)=g(x)
f
(
x
)
h
(
x
)
=
g
(
x
)
, and its
m
n
+
1
mn+1
mn
+
1
coefficients are all positive integers.
algebra
polynomial