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National and Regional Contests
Poland Contests
Polish MO Finals
1980 Polish MO Finals
4
4
Part of
1980 Polish MO Finals
Problems
(1)
every 3 var polymial = equal 2 3 var. polynomials
Source: Polish MO Finals 1980 p4
8/24/2024
Show that for every polynomial
W
W
W
in three variables there exist polynomials
U
U
U
and
V
V
V
such that:
W
(
x
,
y
,
z
)
=
U
(
x
,
y
,
z
)
+
V
(
x
,
y
,
z
)
,
W(x,y,z) = U(x,y,z)+V(x,y,z),
W
(
x
,
y
,
z
)
=
U
(
x
,
y
,
z
)
+
V
(
x
,
y
,
z
)
,
U
(
x
,
y
,
z
)
=
U
(
y
,
x
,
z
)
,
U(x,y,z) = U(y,x,z),
U
(
x
,
y
,
z
)
=
U
(
y
,
x
,
z
)
,
V
(
x
,
y
,
z
)
=
ā
V
(
x
,
z
,
y
)
.
V(x,y,z) = -V(x,z,y).
V
(
x
,
y
,
z
)
=
ā
V
(
x
,
z
,
y
)
.
algebra
polynomial