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Lusophon Mathematical Olympiad
2023 Lusophon Mathematical Olympiad
3
3
Part of
2023 Lusophon Mathematical Olympiad
Problems
(1)
k-special numbers
Source: Lusophon Mathematical Olympiad 2023 Problem 3
5/27/2023
An integer
n
n
n
is called
k
k
k
-special, with
k
k
k
a positive integer, if it's the sum of the squares of
k
k
k
consecutive integers. For example,
13
13
13
is
2
2
2
-special, since
13
=
2
2
+
3
2
13=2^2+3^2
13
=
2
2
+
3
2
, and
2
2
2
is
3
3
3
-special, since
2
=
(
−
1
)
2
+
0
2
+
1
2
2=(-1)^2+0^2+1^2
2
=
(
−
1
)
2
+
0
2
+
1
2
.a) Prove that there's no perfect square that is
4
4
4
-special.b) Find a perfect square that is
I
2
I^2
I
2
-special, for some odd positive integer
I
I
I
with
I
≥
3
I\ge 3
I
≥
3
.
number theory