Rational Points on a Circle

★ 7/10number theory analytic geometry Casework Symmetry

Problem

A point $(x,y)$ is rational if both $x$ and $y$ are rational. For a given positive real $r$ , determine the maximum possible number of rational points on the circle

(x-1)^2+\left(y-\sqrt 2\right)^2=r^2.

Answer

Solution

Difficulty	7/10
Topics	number theory, analytic geometry, Casework, Symmetry

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