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Vietnam National Olympiad
2017 Vietnam National Olympiad
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Vietnam National Olympiad 2017 Problem 6
Vietnam National Olympiad 2017 Problem 6
Source:
January 6, 2017
number theory
Problem Statement
Prove that a)
∑
k
=
1
1008
k
C
2017
k
≡
0
\sum_{k=1}^{1008}kC_{2017}^{k}\equiv 0
∑
k
=
1
1008
k
C
2017
k
≡
0
(mod
201
7
2
2017^2
201
7
2
)b)
∑
k
=
1
504
(
−
1
)
k
C
2017
k
≡
3
(
2
2016
−
1
)
\sum_{k=1}^{504}\left ( -1 \right )^kC_{2017}^{k}\equiv 3\left ( 2^{2016}-1 \right )
∑
k
=
1
504
(
−
1
)
k
C
2017
k
≡
3
(
2
2016
−
1
)
(mod
201
7
2
2017^2
201
7
2
)
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