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Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
1998 Moldova Team Selection Test
8
8
Part of
1998 Moldova Team Selection Test
Problems
(1)
arithmetic progression of maximum length l
Source: Moldova TST 1998
8/8/2023
Let
M
=
{
1
n
∣
n
∈
N
}
M=\{\frac{1}{n}|n\in\mathbb{N}\}
M
=
{
n
1
∣
n
∈
N
}
. Numbers
a
1
,
a
2
,
…
,
a
l
a_1,a_2,\ldots,a_l
a
1
,
a
2
,
…
,
a
l
from an arithmetic progression of maximum length
l
l
l
(
l
≥
3
)
(l\geq 3)
(
l
≥
3
)
if they verify the properties: a) numbers
a
1
,
a
2
,
…
,
a
l
a_1,a_2,\ldots,a_l
a
1
,
a
2
,
…
,
a
l
from a finite arithmetic progression; b) there is no number
b
∈
M
b\in M
b
∈
M
such that numbers
b
,
a
1
,
a
2
,
…
,
a
l
b,a_1,a_2,\ldots,a_l
b
,
a
1
,
a
2
,
…
,
a
l
or
a
1
,
a
2
,
…
,
a
l
,
b
a_1,a_2,\ldots,a_l, b
a
1
,
a
2
,
…
,
a
l
,
b
form a finite arithmetic progression. For example numbers
1
6
,
1
3
,
1
2
∈
M
\frac{1}{6},\frac{1}{3},\frac{1}{2}\in M
6
1
,
3
1
,
2
1
∈
M
form an arithmetic progression of maximum length
3
3
3
. a) FInd an arithmetic progression of maximum length
1998
1998
1998
. b) Prove that there exist maximum arithmetic progressions of any length
l
≥
3
l \geq 3
l
≥
3
.
arithmetic sequence