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National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2005 Korea Junior Math Olympiad
7
7
Part of
2005 Korea Junior Math Olympiad
Problems
(1)
KJMO 2005 n-variable inequality
Source: KJMO 2005 p7
5/1/2019
If positive reals
x
1
,
x
2
,
⋯
,
x
n
x_1,x_2,\cdots,x_n
x
1
,
x
2
,
⋯
,
x
n
satisfy
∑
i
=
1
n
x
i
=
1.
\sum_{i=1}^{n}x_i=1.
∑
i
=
1
n
x
i
=
1.
Prove that
∑
i
=
1
n
1
1
+
∑
j
=
1
i
x
j
<
2
3
∑
i
=
1
n
1
x
i
\sum_{i=1}^{n}\frac{1}{1+\sum_{j=1}^{i}x_j}<\sqrt{\frac{2}{3}\sum_{i=1}^{n}\frac{1}{x_i}}
i
=
1
∑
n
1
+
∑
j
=
1
i
x
j
1
<
3
2
i
=
1
∑
n
x
i
1
algebra
inequalities
Inequality
n-variable inequality