MathDB
Problems
Contests
International Contests
Balkan MO Shortlist
2013 Balkan MO Shortlist
A2
(16ac +a/c^2b+16c/a^2d+4/ac)(bd +b/256d^2c+d/b^2a+1/64bd) >=81/4
(16ac +a/c^2b+16c/a^2d+4/ac)(bd +b/256d^2c+d/b^2a+1/64bd) >=81/4
Source: Balkan MO Shortlist 2013 A2 BMO
March 8, 2020
inequalities
algebra
Problem Statement
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
and
d
d
d
are positive real numbers so that
a
b
c
d
=
1
4
abcd = \frac14
ab
c
d
=
4
1
. Prove that holds
(
16
a
c
+
a
c
2
b
+
16
c
a
2
d
+
4
a
c
)
(
b
d
+
b
256
d
2
c
+
d
b
2
a
+
1
64
b
d
)
≥
81
4
\left( 16ac +\frac{a}{c^2b}+\frac{16c}{a^2d}+\frac{4}{ac}\right)\left( bd +\frac{b}{256d^2c}+\frac{d}{b^2a}+\frac{1}{64bd}\right) \ge \frac{81}{4}
(
16
a
c
+
c
2
b
a
+
a
2
d
16
c
+
a
c
4
)
(
b
d
+
256
d
2
c
b
+
b
2
a
d
+
64
b
d
1
)
≥
4
81
When does the equality hold?
Back to Problems
View on AoPS