MathDB
Problems
Contests
Undergraduate contests
Putnam
2022 Putnam
A3
A3
Part of
2022 Putnam
Problems
(1)
2022 Putnam A3
Source:
12/4/2022
Let
p
p
p
be a prime number greater than 5. Let
f
(
p
)
f(p)
f
(
p
)
denote the number of infinite sequences
a
1
,
a
2
,
a
3
,
…
a_1, a_2, a_3,\ldots
a
1
,
a
2
,
a
3
,
…
such that
a
n
∈
{
1
,
2
,
…
,
p
−
1
}
a_n \in \{1, 2,\ldots, p-1\}
a
n
∈
{
1
,
2
,
…
,
p
−
1
}
and
a
n
a
n
+
2
≡
1
+
a
n
+
1
a_na_{n+2}\equiv1+a_{n+1}
a
n
a
n
+
2
≡
1
+
a
n
+
1
(mod
p
p
p
) for all
n
≥
1.
n\geq 1.
n
≥
1.
Prove that
f
(
p
)
f(p)
f
(
p
)
is congruent to 0 or 2 (mod 5).
Putnam
Putnam 2022