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Putnam
2003 Putnam
4
Putnam 2003 B4
Putnam 2003 B4
Source:
June 23, 2011
Putnam
college contests
Problem Statement
Let
f
(
z
)
=
a
z
4
+
b
z
3
+
c
z
2
+
d
z
+
e
=
a
(
z
−
r
1
)
(
z
−
r
2
)
(
z
−
r
3
)
(
z
−
r
4
)
f(z) = az^4+ bz^3+ cz^2+ dz + e = a(z -r_1)(z -r_2)(z -r_3)(z -r_4)
f
(
z
)
=
a
z
4
+
b
z
3
+
c
z
2
+
d
z
+
e
=
a
(
z
−
r
1
)
(
z
−
r
2
)
(
z
−
r
3
)
(
z
−
r
4
)
where
a
,
b
,
c
,
d
,
e
a, b, c, d, e
a
,
b
,
c
,
d
,
e
are integers,
a
≠
0
a \not= 0
a
=
0
. Show that if
r
1
+
r
2
r_1 + r_2
r
1
+
r
2
is a rational number, and if
r
1
+
r
2
≠
r
3
+
r
4
r_1 + r_2 \neq r_3 + r_4
r
1
+
r
2
=
r
3
+
r
4
, then
r
1
r
2
r_1r_2
r
1
r
2
is a rational number.
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