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Romania National Olympiad
1999 Romania National Olympiad
2
min pos. root of (a+b)x^2-2(ab-1)x-(a+b) = 0
min pos. root of (a+b)x^2-2(ab-1)x-(a+b) = 0
Source: 1999 Romania NMO IX p2
August 15, 2024
algebra
inequalities
Problem Statement
For
a
,
b
>
0
a, b > 0
a
,
b
>
0
, denote by
t
(
a
,
b
)
t(a,b)
t
(
a
,
b
)
the positive root of the equation
(
a
+
b
)
x
2
−
2
(
a
b
−
1
)
x
−
(
a
+
b
)
=
0.
(a+b)x^2-2(ab-1)x-(a+b) = 0.
(
a
+
b
)
x
2
−
2
(
ab
−
1
)
x
−
(
a
+
b
)
=
0.
Let
M
=
{
(
a
.
b
)
∣
a
≠
b
a
n
d
t
(
a
,
b
)
≤
a
b
}
M = \{ (a.b) | \, a \ne b \,\,\, and \,\,\,t(a,b) \le \sqrt{ab} \}
M
=
{(
a
.
b
)
∣
a
=
b
an
d
t
(
a
,
b
)
≤
ab
}
Determine, for
(
a
,
b
)
∈
M
(a, b)\in M
(
a
,
b
)
∈
M
, the mmimum value of
t
(
a
,
b
)
t(a,b)
t
(
a
,
b
)
.
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