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Problems
Contests
International Contests
Austrian-Polish
2005 Austrian-Polish Competition
3
3
Part of
2005 Austrian-Polish Competition
Problems
(1)
Inequality with a(j)-a(i)<=j-i
Source: Austrian-Polish 2005, Problem 3
7/5/2015
Let
a
0
,
a
1
,
a
2
,
.
.
.
,
a
n
a_0, a_1, a_2, ... , a_n
a
0
,
a
1
,
a
2
,
...
,
a
n
be real numbers, which fulfill the following two conditions: a)
0
=
a
0
≤
a
1
≤
a
2
≤
.
.
.
≤
a
n
0 = a_0 \leq a_1 \leq a_2 \leq ... \leq a_n
0
=
a
0
≤
a
1
≤
a
2
≤
...
≤
a
n
. b) For all
0
≤
i
<
j
≤
n
0 \leq i < j \leq n
0
≤
i
<
j
≤
n
holds:
a
j
−
a
i
≤
j
−
i
a_j - a_i \leq j-i
a
j
−
a
i
≤
j
−
i
.Prove that
(
∑
i
=
0
n
a
i
)
2
≥
∑
i
=
0
n
a
i
3
.
\left( \displaystyle \sum_{i=0}^n a_i \right)^2 \geq \sum_{i=0}^n a_i^3.
(
i
=
0
∑
n
a
i
)
2
≥
i
=
0
∑
n
a
i
3
.
inequalities
n-variable inequality