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National and Regional Contests
India Contests
India IMO Training Camp
2011 India IMO Training Camp
3
Two sequences...
Two sequences...
Source: Indian TST Day 4 Problem 3
June 20, 2011
function
number theory unsolved
number theory
Problem Statement
Let
{
a
0
,
a
1
,
…
}
\{a_0,a_1,\ldots\}
{
a
0
,
a
1
,
…
}
and
{
b
0
,
b
1
,
…
}
\{b_0,b_1,\ldots\}
{
b
0
,
b
1
,
…
}
be two infinite sequences of integers such that
(
a
n
−
a
n
−
1
)
(
a
n
−
a
n
−
2
)
+
(
b
n
−
b
n
−
1
)
(
b
n
−
b
n
−
2
)
=
0
(a_{n}-a_{n-1})(a_n-a_{n-2}) +(b_n-b_{n-1})(b_n-b_{n-2})=0
(
a
n
−
a
n
−
1
)
(
a
n
−
a
n
−
2
)
+
(
b
n
−
b
n
−
1
)
(
b
n
−
b
n
−
2
)
=
0
for all integers
n
≥
2
n\geq 2
n
≥
2
. Prove that there exists a positive integer
k
k
k
such that
a
k
+
2011
=
a
k
+
201
1
2011
.
a_{k+2011}=a_{k+2011^{2011}}.
a
k
+
2011
=
a
k
+
201
1
2011
.
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