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Bulgaria Contests
Bulgaria National Olympiad
1996 Bulgaria National Olympiad
1
1
Part of
1996 Bulgaria National Olympiad
Problems
(2)
Bulgaria National Olympiad 1996
Source:
6/10/2020
Find all prime numbers
p
,
q
p,q
p
,
q
for which
p
q
pq
pq
divides
(
5
p
−
2
p
)
(
5
q
−
2
q
)
(5^p-2^p)(5^q-2^q)
(
5
p
−
2
p
)
(
5
q
−
2
q
)
.
number theory
prime numbers
Sequence $\{a_n\}$
Source: Bulgaria 1996
7/28/2020
Sequence
{
a
n
}
\{a_n\}
{
a
n
}
it define
a
1
=
1
a_1=1
a
1
=
1
and
a
n
+
1
=
a
n
n
+
n
a
n
a_{n+1}=\frac{a_n}{n}+\frac{n}{a_n}
a
n
+
1
=
n
a
n
+
a
n
n
for all
n
≥
1
n\ge 1
n
≥
1
\\ Prove that
⌊
a
n
2
⌋
=
n
\lfloor a_n^2\rfloor=n
⌊
a
n
2
⌋
=
n
for all
n
≥
4.
n\ge 4.
n
≥
4.
number theory