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International Contests
APMO
2003 APMO
1
Polynomial
Polynomial
Source: APMO 2003
March 5, 2006
algebra
polynomial
algebra unsolved
Problem Statement
Let
a
,
b
,
c
,
d
,
e
,
f
a,b,c,d,e,f
a
,
b
,
c
,
d
,
e
,
f
be real numbers such that the polynomial
p
(
x
)
=
x
8
−
4
x
7
+
7
x
6
+
a
x
5
+
b
x
4
+
c
x
3
+
d
x
2
+
e
x
+
f
p(x)=x^8-4x^7+7x^6+ax^5+bx^4+cx^3+dx^2+ex+f
p
(
x
)
=
x
8
−
4
x
7
+
7
x
6
+
a
x
5
+
b
x
4
+
c
x
3
+
d
x
2
+
e
x
+
f
factorises into eight linear factors
x
−
x
i
x-x_i
x
−
x
i
, with
x
i
>
0
x_i>0
x
i
>
0
for
i
=
1
,
2
,
…
,
8
i=1,2,\ldots,8
i
=
1
,
2
,
…
,
8
. Determine all possible values of
f
f
f
.
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