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National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2023 Korea Junior Math Olympiad
3
3
Part of
2023 Korea Junior Math Olympiad
Problems
(1)
Sum of squares
Source: KJMO 2023 P3
11/4/2023
Positive integers
a
1
,
a
2
,
…
,
a
2023
a_1, a_2, \dots, a_{2023}
a
1
,
a
2
,
…
,
a
2023
satisfy the following conditions.[*]
a
1
=
5
,
a
2
=
25
a_1 = 5, a_2 = 25
a
1
=
5
,
a
2
=
25
[*]
a
n
+
2
=
7
a
n
+
1
−
a
n
−
6
a_{n + 2} = 7a_{n+1}-a_n-6
a
n
+
2
=
7
a
n
+
1
−
a
n
−
6
for each
n
=
1
,
2
,
…
,
2021
n = 1, 2, \dots, 2021
n
=
1
,
2
,
…
,
2021
Prove that there exist integers
x
,
y
x, y
x
,
y
such that
a
2023
=
x
2
+
y
2
.
a_{2023} = x^2 + y^2.
a
2023
=
x
2
+
y
2
.
number theory
Sequence