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Problems
Contests
National and Regional Contests
Greece Contests
Greece JBMO TST
2017 Greece JBMO TST
1
1
Part of
2017 Greece JBMO TST
Problems
(1)
cyc sum (a+1)\sqrt{2a(1-a)} \geq 8(ab+bc+ca)
Source: Greece JBMO TST 2017, Problem 1
6/25/2018
Positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfy
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Prove that
(
a
+
1
)
2
a
(
1
−
a
)
+
(
b
+
1
)
2
b
(
1
−
b
)
+
(
c
+
1
)
2
c
(
1
−
c
)
≥
8
(
a
b
+
b
c
+
c
a
)
.
(a+1)\sqrt{2a(1-a)} + (b+1)\sqrt{2b(1-b)} + (c+1)\sqrt{2c(1-c)} \geq 8(ab+bc+ca).
(
a
+
1
)
2
a
(
1
−
a
)
+
(
b
+
1
)
2
b
(
1
−
b
)
+
(
c
+
1
)
2
c
(
1
−
c
)
≥
8
(
ab
+
b
c
+
c
a
)
.
Also, find the values of
a
,
b
,
c
a,b,c
a
,
b
,
c
for which the equality happens.
inequalities