MathDB
Problems
Contests
International Contests
Balkan MO
1998 Balkan MO
2
2
Part of
1998 Balkan MO
Problems
(1)
Plus minus inequality by enescu
Source: Balkan MO 1998, Problem 2
4/24/2006
Let
n
≥
2
n\geq 2
n
≥
2
be an integer, and let
0
<
a
1
<
a
2
<
⋯
<
a
2
n
+
1
0 < a_1 < a_2 < \cdots < a_{2n+1}
0
<
a
1
<
a
2
<
⋯
<
a
2
n
+
1
be real numbers. Prove the inequality
a
1
n
−
a
2
n
+
a
3
n
−
⋯
+
a
2
n
+
1
n
<
a
1
−
a
2
+
a
3
−
⋯
+
a
2
n
+
1
n
.
\sqrt[n]{a_1} - \sqrt[n]{a_2} + \sqrt[n]{a_3} - \cdots + \sqrt[n]{a_{2n+1}} < \sqrt[n]{a_1 - a_2 + a_3 - \cdots + a_{2n+1}}.
n
a
1
−
n
a
2
+
n
a
3
−
⋯
+
n
a
2
n
+
1
<
n
a
1
−
a
2
+
a
3
−
⋯
+
a
2
n
+
1
.
Bogdan Enescu, Romania
inequalities
induction
triangle inequality
inequalities proposed