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Taiwan National Olympiad
1994 Taiwan National Olympiad
2
|f(\alpha)-g(\alpha)|
|f(\alpha)-g(\alpha)|
Source: 3-rd Taiwanese Mathematical Olympiad 1994
January 15, 2007
inequalities unsolved
inequalities
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive real numbers and
α
\alpha
α
be any real number. Denote
f
(
α
)
=
a
b
c
(
a
α
+
b
α
+
c
α
)
,
g
(
α
)
=
a
2
+
α
(
b
+
c
−
a
)
+
b
2
+
α
(
−
b
+
c
+
a
)
+
c
2
+
α
(
b
−
c
+
a
)
f(\alpha)=abc(a^{\alpha}+b^{\alpha}+c^{\alpha}), g(\alpha)=a^{2+\alpha}(b+c-a)+b^{2+\alpha}(-b+c+a)+c^{2+\alpha}(b-c+a)
f
(
α
)
=
ab
c
(
a
α
+
b
α
+
c
α
)
,
g
(
α
)
=
a
2
+
α
(
b
+
c
−
a
)
+
b
2
+
α
(
−
b
+
c
+
a
)
+
c
2
+
α
(
b
−
c
+
a
)
. Determine
min
∣
f
(
α
)
−
g
(
α
)
∣
\min{|f(\alpha)-g(\alpha)|}
min
∣
f
(
α
)
−
g
(
α
)
∣
and
max
∣
f
(
α
)
−
g
(
α
)
∣
\max{|f(\alpha)-g(\alpha)|}
max
∣
f
(
α
)
−
g
(
α
)
∣
, if they are exists.
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