MathDB
Problems
Contests
National and Regional Contests
Romania Contests
District Olympiad
2013 District Olympiad
3
Inequality
Inequality
Source: Romania District Olympiad 2013,grade IX(problem 3)
March 10, 2013
inequalities
inequalities proposed
Problem Statement
Let
n
∈
N
∗
n\in {{\mathbb{N}}^{*}}
n
∈
N
∗
and
a
1
,
a
2
,
.
.
.
,
a
n
∈
R
{{a}_{1}},{{a}_{2}},...,{{a}_{n}}\in \mathbb{R}
a
1
,
a
2
,
...
,
a
n
∈
R
so
a
1
+
a
2
+
.
.
.
+
a
k
≤
k
,
(
∀
)
k
∈
{
1
,
2
,
.
.
.
,
n
}
.
{{a}_{1}}+{{a}_{2}}+...+{{a}_{k}}\le k,\left( \forall \right)k\in \left\{ 1,2,...,n \right\}.
a
1
+
a
2
+
...
+
a
k
≤
k
,
(
∀
)
k
∈
{
1
,
2
,
...
,
n
}
.
Prove that
a
1
1
+
a
2
2
+
.
.
.
+
a
n
n
≤
1
1
+
1
2
+
.
.
.
+
1
n
\frac{{{a}_{1}}}{1}+\frac{{{a}_{2}}}{2}+...+\frac{{{a}_{n}}}{n}\le \frac{1}{1}+\frac{1}{2}+...+\frac{1}{n}
1
a
1
+
2
a
2
+
...
+
n
a
n
≤
1
1
+
2
1
+
...
+
n
1
Back to Problems
View on AoPS