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International Contests
IMO Longlists
1979 IMO Longlists
53
53
Part of
1979 IMO Longlists
Problems
(1)
Proving that there exists finitely many sequences...
Source: ILL 1979-53
6/5/2011
An infinite increasing sequence of positive integers
n
j
(
j
=
1
,
2
,
…
)
n_j (j = 1, 2, \ldots )
n
j
(
j
=
1
,
2
,
…
)
has the property that for a certain
c
c
c
,
1
N
∑
n
j
≤
N
n
j
≤
c
,
\frac{1}{N}\sum_{n_j\le N} n_j \le c,
N
1
n
j
≤
N
∑
n
j
≤
c
,
for every
N
>
0
N >0
N
>
0
. Prove that there exist finitely many sequences
m
j
(
i
)
(
i
=
1
,
2
,
…
,
k
)
m^{(i)}_j (i = 1, 2,\ldots, k)
m
j
(
i
)
(
i
=
1
,
2
,
…
,
k
)
such that
{
n
1
,
n
2
,
…
}
=
⋃
i
=
1
k
{
m
1
(
i
)
,
m
2
(
i
)
,
…
}
\{n_1, n_2, \ldots \} =\bigcup_{i=1}^k\{m^{(i)}_1 ,m^{(i)}_2 ,\ldots\}
{
n
1
,
n
2
,
…
}
=
i
=
1
⋃
k
{
m
1
(
i
)
,
m
2
(
i
)
,
…
}
and
m
j
+
1
(
i
)
>
2
m
j
(
i
)
(
1
≤
i
≤
k
,
j
=
1
,
2
,
…
)
.
m^{(i)}_{j+1} > 2m^{(i)}_j (1 \le i \le k, j = 1, 2,\ldots).
m
j
+
1
(
i
)
>
2
m
j
(
i
)
(
1
≤
i
≤
k
,
j
=
1
,
2
,
…
)
.
algebra unsolved
algebra