MathDB
Problems
Contests
National and Regional Contests
Belarus Contests
Belarus Team Selection Test
2024 Belarus Team Selection Test
1.3
1.3
Part of
2024 Belarus Team Selection Test
Problems
(1)
Inequality with reals at least one third
Source: Belarus TST 2024
7/17/2024
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
Inequality
algebra
inequalities