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Baltic Way
2017 Baltic Way
20
20
Part of
2017 Baltic Way
Problems
(1)
Sum of two squares
Source: Baltic Way 2017 Problem 20
11/11/2017
Let
S
S
S
be the set of all ordered pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of integers with
0
<
2
a
<
2
b
<
2017
0<2a<2b<2017
0
<
2
a
<
2
b
<
2017
such that
a
2
+
b
2
a^2+b^2
a
2
+
b
2
is a multiple of
2017
2017
2017
. Prove that
∑
(
a
,
b
)
∈
S
a
=
1
2
∑
(
a
,
b
)
∈
S
b
.
\sum_{(a,b)\in S}a=\frac{1}{2}\sum_{(a,b)\in S}b.
(
a
,
b
)
∈
S
∑
a
=
2
1
(
a
,
b
)
∈
S
∑
b
.
Proposed by Uwe Leck, Germany
number theory