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Problems
Contests
National and Regional Contests
Iran Contests
Pre-Preparation Course Examination
2007 Pre-Preparation Course Examination
13
13
Part of
2007 Pre-Preparation Course Examination
Problems
(1)
The number is not in N-Iran 3rd round-Number Theory 2007
Source:
7/28/2010
Let
{
a
i
}
i
=
1
∞
\{a_i\}_{i=1}^{\infty}
{
a
i
}
i
=
1
∞
be a sequence of positive integers such that
a
1
<
a
2
<
a
3
⋯
a_1<a_2<a_3\cdots
a
1
<
a
2
<
a
3
⋯
and all of primes are members of this sequence. Prove that for every
n
<
m
n<m
n
<
m
1
a
n
+
1
a
n
+
1
+
⋯
+
1
a
m
∉
N
\dfrac{1}{a_n} + \dfrac{1}{a_{n+1}} + \cdots + \dfrac{1}{a_m} \not \in \mathbb N
a
n
1
+
a
n
+
1
1
+
⋯
+
a
m
1
∈
N
number theory unsolved
number theory