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2011 Indonesia MO
6
6
Part of
2011 Indonesia MO
Problems
(1)
2011 divides a_{k-1}a_k - k for all k = 1,2,...,2010
Source: Indonesian Mathematics Olympiad 2011, Day 2, Problem 6
9/14/2011
Let a sequence of integers
a
0
,
a
1
,
a
2
,
⋯
,
a
2010
a_0, a_1, a_2, \cdots, a_{2010}
a
0
,
a
1
,
a
2
,
⋯
,
a
2010
such that
a
0
=
1
a_0 = 1
a
0
=
1
and
2011
2011
2011
divides
a
k
−
1
a
k
−
k
a_{k-1}a_k - k
a
k
−
1
a
k
−
k
for all
k
=
1
,
2
,
⋯
,
2010
k = 1, 2, \cdots, 2010
k
=
1
,
2
,
⋯
,
2010
. Prove that
2011
2011
2011
divides
a
2010
+
1
a_{2010} + 1
a
2010
+
1
.
modular arithmetic
number theory proposed
number theory