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Korea Junior Mathematics Olympiad
2008 Korea Junior Math Olympiad
6
8 \(n^3f_1(n) - 2nf_9(n) + n^2f_3(n)) where, f_s(n) = d_1^s+d_2^s+..+d_k^s
8 \(n^3f_1(n) - 2nf_9(n) + n^2f_3(n)) where, f_s(n) = d_1^s+d_2^s+..+d_k^s
Source: KJMO 2008 p6
May 2, 2019
number theory
Divisors
Sum of powers
Sum
divisible
Problem Statement
If
d
1
,
d
2
,
.
.
.
,
d
k
d_1,d_2,...,d_k
d
1
,
d
2
,
...
,
d
k
are all distinct positive divisors of
n
n
n
, we defi ne
f
s
(
n
)
=
d
1
s
+
d
2
s
+
.
.
+
d
k
s
f_s(n) = d_1^s+d_2^s+..+d_k^s
f
s
(
n
)
=
d
1
s
+
d
2
s
+
..
+
d
k
s
. For example, we have
f
1
(
3
)
=
1
+
3
=
4
,
f
2
(
4
)
=
1
+
2
2
+
4
2
=
21
f_1(3) = 1 + 3 = 4, f_2(4) = 1 + 2^2 + 4^2 = 21
f
1
(
3
)
=
1
+
3
=
4
,
f
2
(
4
)
=
1
+
2
2
+
4
2
=
21
. Prove that for all positive integers
n
n
n
,
n
3
f
1
(
n
)
−
2
n
f
9
(
n
)
+
n
2
f
3
(
n
)
n^3f_1(n) - 2nf_9(n) + n^2f_3(n)
n
3
f
1
(
n
)
−
2
n
f
9
(
n
)
+
n
2
f
3
(
n
)
is divisible by
8
8
8
.
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