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National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2023 Junior Balkan Team Selection Tests - Moldova
12
\frac{a^4+3ab^3}{a^3+2b^3}+\frac{b^4+3bc^3}{b^3+2c^3}+\frac{c^4+3ca^3}{c^3+2a^3}
\frac{a^4+3ab^3}{a^3+2b^3}+\frac{b^4+3bc^3}{b^3+2c^3}+\frac{c^4+3ca^3}{c^3+2a^3}
Source: Moldova JTST 2023
May 5, 2023
inequalities
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers such that
a
2
+
b
2
+
c
2
=
3.
a^2+b^2+c^2=3.
a
2
+
b
2
+
c
2
=
3.
Prove that
a
4
+
3
a
b
3
a
3
+
2
b
3
+
b
4
+
3
b
c
3
b
3
+
2
c
3
+
c
4
+
3
c
a
3
c
3
+
2
a
3
≤
4.
\frac{a^4+3ab^3}{a^3+2b^3}+\frac{b^4+3bc^3}{b^3+2c^3}+\frac{c^4+3ca^3}{c^3+2a^3}\leq4.
a
3
+
2
b
3
a
4
+
3
a
b
3
+
b
3
+
2
c
3
b
4
+
3
b
c
3
+
c
3
+
2
a
3
c
4
+
3
c
a
3
≤
4.
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