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Regional Mathematical Olympiad
2005 India Regional Mathematical Olympiad
7
Quadratic returns!
Quadratic returns!
Source: Indian RMO 2005 Problem 7
February 28, 2006
quadratics
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be three positive real numbers such that
a
+
b
+
c
=
1
a+ b +c =1
a
+
b
+
c
=
1
. Let
λ
=
m
i
n
{
a
3
+
a
2
b
c
,
b
3
+
b
2
a
c
,
c
3
+
a
b
c
2
}
\lambda = min \{ a^3 + a^2bc , b^3 + b^2 ac , c^3 + ab c^2 \}
λ
=
min
{
a
3
+
a
2
b
c
,
b
3
+
b
2
a
c
,
c
3
+
ab
c
2
}
Prove that the roots of
x
2
+
x
+
4
λ
=
0
x^2 + x + 4 \lambda = 0
x
2
+
x
+
4
λ
=
0
are real.
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