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2007 Moldova National Olympiad
11.5
Moldova NMO, 11th grade, Day 2, Problem 5
Moldova NMO, 11th grade, Day 2, Problem 5
Source: Rather Easy
March 4, 2007
inequalities
induction
function
inequalities unsolved
Problem Statement
Real numbers
a
1
,
a
2
,
…
,
a
n
a_{1},a_{2},\dots,a_{n}
a
1
,
a
2
,
…
,
a
n
satisfy
a
i
≥
1
i
a_{i}\geq\frac{1}{i}
a
i
≥
i
1
, for all
i
=
1
,
n
‾
i=\overline{1,n}
i
=
1
,
n
. Prove the inequality:
(
a
1
+
1
)
(
a
2
+
1
2
)
⋅
⋯
⋅
(
a
n
+
1
n
)
≥
2
n
(
n
+
1
)
!
(
1
+
a
1
+
2
a
2
+
⋯
+
n
a
n
)
.
\left(a_{1}+1\right)\left(a_{2}+\frac{1}{2}\right)\cdot\dots\cdot\left(a_{n}+\frac{1}{n}\right)\geq\frac{2^{n}}{(n+1)!}(1+a_{1}+2a_{2}+\dots+na_{n}).
(
a
1
+
1
)
(
a
2
+
2
1
)
⋅
⋯
⋅
(
a
n
+
n
1
)
≥
(
n
+
1
)!
2
n
(
1
+
a
1
+
2
a
2
+
⋯
+
n
a
n
)
.
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