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2011 Bulgaria National Olympiad
1
Binomial diophantine equation
Binomial diophantine equation
Source: Bulgaria MO 2011
May 30, 2011
number theory proposed
number theory
Problem Statement
Prove whether or not there exist natural numbers
n
,
k
n,k
n
,
k
where
1
≤
k
≤
n
−
2
1\le k\le n-2
1
≤
k
≤
n
−
2
such that
(
n
k
)
2
+
(
n
k
+
1
)
2
=
(
n
k
+
2
)
4
\binom{n}{k}^2+\binom{n}{k+1}^2=\binom{n}{k+2}^4
(
k
n
)
2
+
(
k
+
1
n
)
2
=
(
k
+
2
n
)
4
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