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National and Regional Contests
Thailand Contests
Thailand National Olympiad
2015 Thailand Mathematical Olympiad
2
2
Part of
2015 Thailand Mathematical Olympiad
Problems
(1)
sum a^5/(a^3 + 1) >= 3/2 if abc=1, a,b,c>0
Source: Thailand Mathematical Olympiad 2015 p2
8/16/2020
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive reals with
a
b
c
=
1
abc = 1
ab
c
=
1
. Prove the inequality
a
5
a
3
+
1
+
b
5
b
3
+
1
+
c
5
c
3
+
1
≥
3
2
\frac{a^5}{a^3 + 1}+\frac{b^5}{b^3 + 1}+\frac{c^5}{c^3 + 1} \ge \frac32
a
3
+
1
a
5
+
b
3
+
1
b
5
+
c
3
+
1
c
5
≥
2
3
and determine all values of a, b, c for which equality is attained
algebra
inequalities