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Korea Junior Mathematics Olympiad
2024 Korea Junior Math Olympiad (First Round)
14.
14.
Part of
2024 Korea Junior Math Olympiad (First Round)
Problems
(1)
Algebra with numbers
Source: 2024 KJMO first round
10/26/2024
Find the number of positive integer
x
x
x
that has
a
1
,
a
2
,
⋅
⋅
⋅
a
20
{a}_{1},{a}_{2},\cdot \cdot \cdot {a}_{20}
a
1
,
a
2
,
⋅
⋅
⋅
a
20
which follows the following (
x
≥
1000
x \ge 1000
x
≥
1000
)1)
a
1
=
2
,
a
2
=
1
,
a
3
=
x
{a}_{1}=2, {a}_{2}=1, {a}_{3}=x
a
1
=
2
,
a
2
=
1
,
a
3
=
x
2) for positive integer
n
n
n
, (
4
≤
n
≤
20
4 \le n \le 20
4
≤
n
≤
20
),
a
n
=
a
n
−
3
+
(
−
2
)
n
a
n
−
1
a
n
−
2
{a}_{n}={a}_{n-3}+\frac{(-2)^n}{{a}_{n-1}{a}_{n-2}}
a
n
=
a
n
−
3
+
a
n
−
1
a
n
−
2
(
−
2
)
n
algebra
number
NumberTheory