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National and Regional Contests
France Contests
France Team Selection Test
2000 France Team Selection Test
3
France TST 2000 D1Q3
France TST 2000 D1Q3
Source:
March 2, 2013
inequalities
inequalities unsolved
Problem Statement
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
are positive reals with sum
1
1
1
. Show that
a
2
a
+
b
+
b
2
b
+
c
+
c
2
c
+
d
+
d
2
d
+
a
≥
1
2
\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+d}+\frac{d^2}{d+a} \ge \frac{1}{2}
a
+
b
a
2
+
b
+
c
b
2
+
c
+
d
c
2
+
d
+
a
d
2
≥
2
1
with equality iff
a
=
b
=
c
=
d
=
1
4
a=b=c=d=\frac{1}{4}
a
=
b
=
c
=
d
=
4
1
.
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