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1.3
Inequality with reals at least one third
Inequality with reals at least one third
Source: Belarus TST 2024
July 17, 2024
Inequality
algebra
inequalities
Problem Statement
Prove that for any real numbers
a
,
b
,
c
,
d
≥
1
3
a,b,c,d \geq \frac{1}{3}
a
,
b
,
c
,
d
≥
3
1
the following inequality holds:
a
6
b
4
+
c
3
+
b
6
c
4
+
d
3
+
c
6
d
4
+
a
3
+
d
6
a
4
+
b
3
≥
a
+
b
+
c
+
d
4
\sqrt{\frac{a^6}{b^4+c^3}+\frac{b^6}{c^4+d^3}+\frac{c^6}{d^4+a^3}+\frac{d^6}{a^4+b^3}}\geq \frac{a+b+c+d}{4}
b
4
+
c
3
a
6
+
c
4
+
d
3
b
6
+
d
4
+
a
3
c
6
+
a
4
+
b
3
d
6
≥
4
a
+
b
+
c
+
d
D. Zmiaikou
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