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International Contests
Austrian-Polish
1992 Austrian-Polish Competition
4
4
Part of
1992 Austrian-Polish Competition
Problems
(1)
P(x) = (x - u^k) (x - uv) (x -v^k) = x^3 + ax^2 + bx + c
Source: Austrian - Polish 1992 APMC
5/7/2020
Let
k
k
k
be a positive integer and
u
,
v
u, v
u
,
v
be real numbers. Consider
P
(
x
)
=
(
x
−
u
k
)
(
x
−
u
v
)
(
x
−
v
k
)
=
x
3
+
a
x
2
+
b
x
+
c
P(x) = (x - u^k) (x - uv) (x -v^k) = x^3 + ax^2 + bx + c
P
(
x
)
=
(
x
−
u
k
)
(
x
−
uv
)
(
x
−
v
k
)
=
x
3
+
a
x
2
+
b
x
+
c
. (a) For
k
=
2
k = 2
k
=
2
prove that if
a
,
b
,
c
a, b, c
a
,
b
,
c
are rational then so is
u
v
uv
uv
. (b) Is that also true for
k
=
3
k = 3
k
=
3
?
polynomial
algebra