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Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1999 Poland - Second Round
6
6
Part of
1999 Poland - Second Round
Problems
(1)
a_1+2^ka_2+3^ka_3+...+n^ka_n is divisible by k!
Source: Polish second round 1999 p6
1/19/2020
Suppose that
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
are integers such that
a
1
+
2
i
a
2
+
3
i
a
3
+
.
.
.
+
n
i
a
n
=
0
a_1 +2^ia_2 +3^ia_3 +...+n^ia_n = 0
a
1
+
2
i
a
2
+
3
i
a
3
+
...
+
n
i
a
n
=
0
for
i
=
1
,
2
,
.
.
.
,
k
−
1
i = 1,2,...,k -1
i
=
1
,
2
,
...
,
k
−
1
, where
k
≥
2
k \ge 2
k
≥
2
is a given integer. Prove that
a
1
+
2
k
a
2
+
3
k
a
3
+
.
.
.
+
n
k
a
n
a_1+2^ka_2+3^ka_3+...+n^ka_n
a
1
+
2
k
a
2
+
3
k
a
3
+
...
+
n
k
a
n
is divisible by
k
!
k!
k
!
.
factorial
number theory
Sum