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Bulgaria Contests
Bulgaria National Olympiad
1978 Bulgaria National Olympiad
Problem 1
sequence is in N
sequence is in N
Source: Bulgaria 1978 P1
June 17, 2021
Sequences
number theory
Problem Statement
We are given the sequence
a
1
,
a
2
,
a
3
,
…
a_1,a_2,a_3,\ldots
a
1
,
a
2
,
a
3
,
…
, for which:
a
n
=
a
n
−
1
2
+
c
a
n
−
2
for all
n
>
2.
a_n=\frac{a^2_{n-1}+c}{a_{n-2}}\enspace\text{for all }n>2.
a
n
=
a
n
−
2
a
n
−
1
2
+
c
for all
n
>
2.
Prove that the numbers
a
1
a_1
a
1
,
a
2
a_2
a
2
and
a
1
2
+
a
2
2
+
c
a
1
a
2
\frac{a_1^2+a_2^2+c}{a_1a_2}
a
1
a
2
a
1
2
+
a
2
2
+
c
are whole numbers.
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