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National and Regional Contests
Russia Contests
Moscow Mathematical Olympiad
1954 Moscow Mathematical Olympiad
267
267
Part of
1954 Moscow Mathematical Olympiad
Problems
(1)
MMO 267 Moscow MO 1954 (x - x_0)^2/ (x^4+a_1x^3+a_2x^2+a_3x+a_4)
Source:
8/13/2019
Prove that if
x
0
4
+
a
1
x
0
3
+
a
2
x
0
2
+
a
3
x
0
+
a
4
=
0
a
n
d
4
x
0
3
+
3
a
1
x
0
2
+
2
a
2
x
0
+
a
3
=
0
,
x^4_0+ a_1x^3_0+ a_2x^2_0+ a_3x_0 + a_4 = 0 \ \ and \ \ 4x^3_0+ 3a_1x^2_0+ 2a_2x_0 + a_3 = 0,
x
0
4
+
a
1
x
0
3
+
a
2
x
0
2
+
a
3
x
0
+
a
4
=
0
an
d
4
x
0
3
+
3
a
1
x
0
2
+
2
a
2
x
0
+
a
3
=
0
,
then
x
4
+
a
1
x
3
+
a
2
x
2
+
a
3
x
+
a
4
x^4 + a_1x^3 + a_2x^2 + a_3x + a_4
x
4
+
a
1
x
3
+
a
2
x
2
+
a
3
x
+
a
4
is a mutliple of
(
x
−
x
0
)
2
(x - x_0)^2
(
x
−
x
0
)
2
.
polynomial
divisible
Divide
algebra