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National and Regional Contests
Poland Contests
Poland - Second Round
1975 Poland - Second Round
1
1
Part of
1975 Poland - Second Round
Problems
(1)
W(x) = x^4 + ax^3 + bx + cx + d
Source: Polish MO Second Round 1975 p1
9/8/2024
The polynomial
W
(
x
)
=
x
4
+
a
x
3
+
b
x
+
c
x
+
d
W(x) = x^4 + ax^3 + bx + cx + d
W
(
x
)
=
x
4
+
a
x
3
+
b
x
+
c
x
+
d
is given. Prove that if the equation
W
(
x
)
=
0
W(x) = 0
W
(
x
)
=
0
has four real roots, then for there to exist
m
m
m
such that
W
(
x
+
m
)
=
x
4
+
p
x
2
+
q
W(x+m) = x^4+px^2+q
W
(
x
+
m
)
=
x
4
+
p
x
2
+
q
, it is necessary and it is enough that the sum of certain two roots of the equation
W
(
x
)
=
0
W(x) = 0
W
(
x
)
=
0
equals the sum of the remaining ones.
algebra
polynomial