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IMC 2018 · Problem 10

Day 225th IMC · Blagoevgrad, Bulgaria

Statement

For R>1R > 1 let DR={(a,b)∈Z2:0<a2+b2<R}\mathcal{D}_R = \{(a, b) \in \mathbb{Z}^2 : 0 < a^2 + b^2 < R\}. Compute

lim⁡R→∞∑(a,b)∈DR(−1)a+ba2+b2.\lim_{R \to \infty} \sum_{(a,b) \in \mathcal{D}_R} \frac{(-1)^{a+b}}{a^2 + b^2}.

Official solution

Hidden so you can work on the problem first.

Proposed by Rodrigo Angelo, Princeton University and Matheus Secco, PUC, Rio de Janeiro.