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National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1996 Vietnam National Olympiad
3
VietNam MO 1996
VietNam MO 1996
Source:
February 18, 2007
algebra
polynomial
calculus
derivative
inequalities unsolved
inequalities
Problem Statement
Prove that:
a
+
b
+
c
+
d
≥
2
3
(
a
b
+
b
c
+
c
a
+
a
d
+
a
c
+
b
d
)
a+b+c+d \geq \frac{2}{3}(ab+bc+ca+ad+ac+bd)
a
+
b
+
c
+
d
≥
3
2
(
ab
+
b
c
+
c
a
+
a
d
+
a
c
+
b
d
)
where
a
;
b
;
c
;
d
a;b;c;d
a
;
b
;
c
;
d
are positive real numbers satisfying
2
(
a
b
+
b
c
+
c
d
+
d
a
+
a
c
+
b
d
)
+
a
b
c
+
b
c
d
+
c
d
a
+
d
a
b
=
16
2(ab+bc+cd+da+ac+bd)+abc+bcd+cda+dab=16
2
(
ab
+
b
c
+
c
d
+
d
a
+
a
c
+
b
d
)
+
ab
c
+
b
c
d
+
c
d
a
+
d
ab
=
16
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