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Taiwan APMO Prelininary
2020 Taiwan APMO Preliminary
P4
P4
Part of
2020 Taiwan APMO Preliminary
Problems
(1)
2020 Taiwan APMO Preliminary Problem 4
Source: 2020 Taiwan APMO Preliminary
7/23/2020
Let
(
a
,
b
)
=
(
a
n
,
a
n
+
1
)
,
∀
n
∈
N
(a,b)=(a_n,a_{n+1}),\forall n\in\mathbb{N}
(
a
,
b
)
=
(
a
n
,
a
n
+
1
)
,
∀
n
∈
N
all be positive interger solutions that satisfies
1
≤
a
≤
b
1\leq a\leq b
1
≤
a
≤
b
and
a
2
+
b
2
+
a
+
b
+
1
a
b
∈
N
\dfrac{a^2+b^2+a+b+1}{ab}\in\mathbb{N}
ab
a
2
+
b
2
+
a
+
b
+
1
∈
N
And the value of
a
n
a_n
a
n
is only determined by the following recurrence relation:
a
n
+
2
=
p
a
n
+
1
+
q
a
n
+
r
a_{n+2} = pa_{n+1} + qa_n + r
a
n
+
2
=
p
a
n
+
1
+
q
a
n
+
r
Find
(
p
,
q
,
r
)
(p,q,r)
(
p
,
q
,
r
)
.
algebra
number theory