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National and Regional Contests
PEN Problems
PEN G Problems
29
G 29
G 29
Source:
May 25, 2007
algebra
polynomial
quadratics
Irrational numbers
Problem Statement
Let
p
(
x
)
=
x
3
+
a
1
x
2
+
a
2
x
+
a
3
p(x)=x^{3}+a_{1}x^{2}+a_{2}x+a_{3}
p
(
x
)
=
x
3
+
a
1
x
2
+
a
2
x
+
a
3
have rational coefficients and have roots
r
1
r_{1}
r
1
,
r
2
r_{2}
r
2
, and
r
3
r_{3}
r
3
. If
r
1
−
r
2
r_{1}-r_{2}
r
1
−
r
2
is rational, must
r
1
r_{1}
r
1
,
r
2
r_{2}
r
2
, and
r
3
r_{3}
r
3
be rational?
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