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1985 IMO Longlists
14
IMO LongList 1985 CAN3 - Prove the divisibility
IMO LongList 1985 CAN3 - Prove the divisibility
Source:
September 10, 2010
number theory unsolved
number theory
Problem Statement
Let
k
k
k
be a positive integer. Define
u
0
=
0
,
u
1
=
1
u_0 = 0, u_1 = 1
u
0
=
0
,
u
1
=
1
, and
u
n
=
k
u
n
−
1
−
u
n
−
2
,
n
≥
2.
u_n=ku_{n-1}-u_{n-2} , n \geq 2.
u
n
=
k
u
n
−
1
−
u
n
−
2
,
n
≥
2.
Show that for each integer
n
n
n
, the number
u
1
3
+
u
2
3
+
⋯
+
u
n
3
u_1^3 + u_2^3 +\cdots+ u_n^3
u
1
3
+
u
2
3
+
⋯
+
u
n
3
is a multiple of
u
1
+
u
2
+
⋯
+
u
n
.
u_1 + u_2 +\cdots+ u_n.
u
1
+
u
2
+
⋯
+
u
n
.
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