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Tuymaada Olympiad
2014 Tuymaada Olympiad
3
3
Part of
2014 Tuymaada Olympiad
Problems
(1)
Sum of 1/\sqrt{x^3+1} when sum of 1/x=3
Source: Tuymaada 2014, Day 1, Problem 4 Juniors, Problem 3 Seniors
7/12/2014
Positive numbers
a
,
b
,
c
a,\ b,\ c
a
,
b
,
c
satisfy
1
a
+
1
b
+
1
c
=
3
\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=3
a
1
+
b
1
+
c
1
=
3
. Prove the inequality
1
a
3
+
1
+
1
b
3
+
1
+
1
c
3
+
1
≤
3
2
.
\dfrac{1}{\sqrt{a^3+1}}+\dfrac{1}{\sqrt{b^3+1}}+\dfrac{1}{\sqrt{c^3+1}}\le \dfrac{3}{\sqrt{2}}.
a
3
+
1
1
+
b
3
+
1
1
+
c
3
+
1
1
≤
2
3
.
(N. Alexandrov)
inequalities
function
Tuymaada