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2010 IMO Shortlist
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8
Part of
2010 IMO Shortlist
Problems
(1)
South Korean IMO Shortlist 2010 Inequality
Source: IMO Shortlist 2010, Algebra 8
7/17/2011
Given six positive numbers
a
,
b
,
c
,
d
,
e
,
f
a,b,c,d,e,f
a
,
b
,
c
,
d
,
e
,
f
such that
a
<
b
<
c
<
d
<
e
<
f
.
a < b < c < d < e < f.
a
<
b
<
c
<
d
<
e
<
f
.
Let
a
+
c
+
e
=
S
a+c+e=S
a
+
c
+
e
=
S
and
b
+
d
+
f
=
T
.
b+d+f=T.
b
+
d
+
f
=
T
.
Prove that
2
S
T
>
3
(
S
+
T
)
(
S
(
b
d
+
b
f
+
d
f
)
+
T
(
a
c
+
a
e
+
c
e
)
)
.
2ST > \sqrt{3(S+T)\left(S(bd + bf + df) + T(ac + ae + ce) \right)}.
2
ST
>
3
(
S
+
T
)
(
S
(
b
d
+
b
f
+
df
)
+
T
(
a
c
+
a
e
+
ce
)
)
.
Proposed by Sung Yun Kim, South Korea
inequalities
algebra
IMO Shortlist