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4
Another year-based olympiad problem
Another year-based olympiad problem
Source: 2016 BAMO-12 #4
February 25, 2016
algebra
Proof
Olympiad
Problem Statement
Find a positive integer
N
N
N
and
a
1
,
a
2
,
⋯
,
a
N
a_1, a_2, \cdots, a_N
a
1
,
a
2
,
⋯
,
a
N
where
a
k
=
1
a_k = 1
a
k
=
1
or
a
k
=
−
1
a_k = -1
a
k
=
−
1
, for each
k
=
1
,
2
,
⋯
,
N
,
k=1,2,\cdots,N,
k
=
1
,
2
,
⋯
,
N
,
such that
a
1
⋅
1
3
+
a
2
⋅
2
3
+
a
3
⋅
3
3
⋯
+
a
N
⋅
N
3
=
20162016
a_1 \cdot 1^3 + a_2 \cdot 2^3 + a_3 \cdot 3^3 \cdots + a_N \cdot N^3 = 20162016
a
1
⋅
1
3
+
a
2
⋅
2
3
+
a
3
⋅
3
3
⋯
+
a
N
⋅
N
3
=
20162016
or show that this is impossible.
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