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Mongolia Contests
Mongolian Mathematical Olympiad
2000 Mongolian Mathematical Olympiad
Problem 5
inequality in positive integers
inequality in positive integers
Source: Mongolia MO 2000 Teachers P5
April 22, 2021
inequalities
number theory
Problem Statement
Let
m
,
n
,
k
m,n,k
m
,
n
,
k
be positive integers with
m
≥
2
m\ge2
m
≥
2
and
k
≥
log
2
(
m
−
1
)
k\ge\log_2(m-1)
k
≥
lo
g
2
(
m
−
1
)
. Prove that
∏
s
=
1
n
m
s
−
1
m
s
<
1
2
n
+
1
2
k
+
1
.
\prod_{s=1}^n\frac{ms-1}{ms}<\sqrt[2^{k+1}]{\frac1{2n+1}}.
s
=
1
∏
n
m
s
m
s
−
1
<
2
k
+
1
2
n
+
1
1
.
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