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Balkan MO Shortlist
2018 Balkan MO Shortlist
A3
Inequality with positive integer
Inequality with positive integer
Source: Shortlist BMO 2018, A3
May 2, 2019
inequalities
Problem Statement
Show that for every positive integer
n
n
n
we have:
∑
k
=
0
n
(
2
n
+
1
−
k
k
+
1
)
k
=
(
2
n
+
1
1
)
0
+
(
2
n
2
)
1
+
.
.
.
+
(
n
+
1
n
+
1
)
n
≤
2
n
\sum_{k=0}^{n}\left(\frac{2n+1-k}{k+1}\right)^k=\left(\frac{2n+1}{1}\right)^0+\left(\frac{2n}{2}\right)^1+...+\left(\frac{n+1}{n+1}\right)^n\leq 2^n
k
=
0
∑
n
(
k
+
1
2
n
+
1
−
k
)
k
=
(
1
2
n
+
1
)
0
+
(
2
2
n
)
1
+
...
+
(
n
+
1
n
+
1
)
n
≤
2
n
Proposed by Dorlir Ahmeti, Albania
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