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1998 Romania National Olympiad
1
x + y = a, x^3 + y^3 >= a
x + y = a, x^3 + y^3 >= a
Source: 1998 Romania NMO VIII p1
August 14, 2024
algebra
system of equations
inequalities
Problem Statement
Let
a
a
a
be a real number and
A
=
{
(
x
,
y
)
∈
R
×
R
∣
x
+
y
=
a
}
A = \{(x, y) \in R \times R | \, x + y = a\}
A
=
{(
x
,
y
)
∈
R
×
R
∣
x
+
y
=
a
}
,
B
=
{
(
x
,
y
)
∈
R
×
R
∣
x
3
+
y
3
<
a
}
B = \{(x,y) \in R \times R | \, x^3 + y^3 < a\}
B
=
{(
x
,
y
)
∈
R
×
R
∣
x
3
+
y
3
<
a
}
. Find all values of
a
a
a
such that
A
∩
B
=
∅
A \cap B = \emptyset
A
∩
B
=
∅
.
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