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Lithuania Contests
Lithuania Team Selection Test
2006 Lithuania Team Selection Test
1
Quite well known and easy
Quite well known and easy
Source: Lithuanian TST 2006
April 21, 2006
inequalities
n-variable inequality
Problem Statement
Let
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \dots, a_n
a
1
,
a
2
,
…
,
a
n
be positive real numbers, whose sum is
1
1
1
. Prove that
a
1
2
a
1
+
a
2
+
a
2
2
a
2
+
a
3
+
⋯
+
a
n
−
1
2
a
n
−
1
+
a
n
+
a
n
2
a
n
+
a
1
≥
1
2
\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+\dots+\frac{a_{n-1}^2}{a_{n-1}+a_n}+\frac{a_n^2}{a_n+a_1}\ge \frac{1}{2}
a
1
+
a
2
a
1
2
+
a
2
+
a
3
a
2
2
+
⋯
+
a
n
−
1
+
a
n
a
n
−
1
2
+
a
n
+
a
1
a
n
2
≥
2
1
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