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National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2017 Korea Junior Math Olympiad
3
3
Part of
2017 Korea Junior Math Olympiad
Problems
(1)
Divisors of 15^25+1
Source: KJMO 2017 p3
7/26/2019
Find all
n
>
1
n>1
n
>
1
and integers
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\dots,a_n
a
1
,
a
2
,
…
,
a
n
satisfying the following three conditions: (i)
2
<
a
1
≤
a
2
≤
⋯
≤
a
n
2<a_1\le a_2\le \cdots\le a_n
2
<
a
1
≤
a
2
≤
⋯
≤
a
n
(ii)
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\dots,a_n
a
1
,
a
2
,
…
,
a
n
are divisors of
1
5
25
+
1
15^{25}+1
1
5
25
+
1
. (iii)
2
−
2
1
5
25
+
1
=
(
1
−
2
a
1
)
+
(
1
−
2
a
2
)
+
⋯
+
(
1
−
2
a
n
)
2-\frac{2}{15^{25}+1}=\left(1-\frac{2}{a_1}\right)+\left(1-\frac{2}{a_2}\right)+\cdots+\left(1-\frac{2}{a_n}\right)
2
−
1
5
25
+
1
2
=
(
1
−
a
1
2
)
+
(
1
−
a
2
2
)
+
⋯
+
(
1
−
a
n
2
)
algebra
number theory
KJMO