MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2007 Junior Balkan Team Selection Tests - Moldova
2
2
Part of
2007 Junior Balkan Team Selection Tests - Moldova
Problems
(1)
sum 1/(a_1+ a_2) >1 if a_i^2 /(a_i-1)>S, a_>1, sum a_i=S
Source: 2007 Moldova JBMO TST p2
2/20/2021
The real numbers
a
1
,
a
2
,
a
3
a_1, a_2, a_3
a
1
,
a
2
,
a
3
are greater than
1
1
1
and have the sum equal to
S
S
S
. If for any
i
=
1
,
2
,
3
i = 1, 2, 3
i
=
1
,
2
,
3
, holds the inequality
a
i
2
a
i
−
1
>
S
\frac{a_i^2}{a_i-1}>S
a
i
−
1
a
i
2
>
S
, prove the inequality
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
algebra
inequalities