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National and Regional Contests
France Contests
France Team Selection Test
2007 France Team Selection Test
2
France TST 2007
France TST 2007
Source: Problem 2
May 16, 2007
inequalities
search
symmetry
inequalities proposed
Problem Statement
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be positive reals such taht
a
+
b
+
c
+
d
=
1
a+b+c+d=1
a
+
b
+
c
+
d
=
1
. Prove that:
6
(
a
3
+
b
3
+
c
3
+
d
3
)
≥
a
2
+
b
2
+
c
2
+
d
2
+
1
8
.
6(a^{3}+b^{3}+c^{3}+d^{3})\geq a^{2}+b^{2}+c^{2}+d^{2}+\frac{1}{8}.
6
(
a
3
+
b
3
+
c
3
+
d
3
)
≥
a
2
+
b
2
+
c
2
+
d
2
+
8
1
.
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