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All-Russian Olympiad
2005 All-Russian Olympiad
3
ARO 2005 inequality
ARO 2005 inequality
Source: ARO 2005 - problem 9.3
April 30, 2005
inequalities
algebra unsolved
algebra
Problem Statement
Given three reals
a
1
,
a
2
,
a
3
>
1
,
S
=
a
1
+
a
2
+
a
3
a_1,\,a_2,\,a_3>1,\,S=a_1+a_2+a_3
a
1
,
a
2
,
a
3
>
1
,
S
=
a
1
+
a
2
+
a
3
. Provided
a
i
2
a
i
−
1
>
S
{a_i^2\over a_i-1}>S
a
i
−
1
a
i
2
>
S
for every
i
=
1
,
2
,
3
i=1,\,2,\,3
i
=
1
,
2
,
3
prove that
1
a
1
+
a
2
+
1
a
2
+
a
3
+
1
a
3
+
a
1
>
1.
\frac{1}{a_1+a_2}+\frac{1}{a_2+a_3}+\frac{1}{a_3+a_1}>1.
a
1
+
a
2
1
+
a
2
+
a
3
1
+
a
3
+
a
1
1
>
1.
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