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IMC
2018 IMC
10
IMC 2018 P10
IMC 2018 P10
Source: IMC 2018 P10
July 25, 2018
college contests
imc2018
real analysis
Problem Statement
For
R
>
1
R>1
R
>
1
let
D
R
=
{
(
a
,
b
)
∈
Z
2
:
0
<
a
2
+
b
2
<
R
}
\mathcal{D}_R =\{ (a,b)\in \mathbb{Z}^2: 0<a^2+b^2<R\}
D
R
=
{(
a
,
b
)
∈
Z
2
:
0
<
a
2
+
b
2
<
R
}
. Compute
lim
R
→
∞
∑
(
a
,
b
)
∈
D
R
(
−
1
)
a
+
b
a
2
+
b
2
.
\lim_{R\rightarrow \infty}{\sum_{(a,b)\in \mathcal{D}_R}{\frac{(-1)^{a+b}}{a^2+b^2}}}.
R
→
∞
lim
(
a
,
b
)
∈
D
R
∑
a
2
+
b
2
(
−
1
)
a
+
b
.
Proposed by Rodrigo Angelo, Princeton University and Matheus Secco, PUC, Rio de Janeiro
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