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2000 Moldova National Olympiad
Problem 3
show number is prime, {1},{2,3,4},{5,6,7,8,9}
show number is prime, {1},{2,3,4},{5,6,7,8,9}
Source: Moldova MO 2000 Grade 7 P3
April 23, 2021
number theory
Problem Statement
Consider the sets
A
1
=
{
1
}
A_1=\{1\}
A
1
=
{
1
}
,
A
2
=
{
2
,
3
,
4
}
A_2=\{2,3,4\}
A
2
=
{
2
,
3
,
4
}
,
A
3
=
{
5
,
6
,
7
,
8
,
9
}
A_3=\{5,6,7,8,9\}
A
3
=
{
5
,
6
,
7
,
8
,
9
}
, etc. Let
b
n
b_n
b
n
be the arithmetic mean of the smallest and the greatest element in
A
n
A_n
A
n
. Show that the number
2000
b
1
−
1
+
2000
b
2
−
1
+
…
+
2000
b
2000
−
1
\frac{2000}{b_1-1}+\frac{2000}{b_2-1}+\ldots+\frac{2000}{b_{2000}-1}
b
1
−
1
2000
+
b
2
−
1
2000
+
…
+
b
2000
−
1
2000
is a prime integer.
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