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rational points on circles

Source: LIMIT 2019 CBS2 P12

April 28, 2021
number theory

Problem Statement

Find the number of rational solutions of the following equations (i.e., rational xx and yy satisfy the equations) x2+y2=2x^2+y^2=2x2+y2=3x^2+y^2=3<spanclass=′latex−bold′>(A)</span> 2 and 2<span class='latex-bold'>(A)</span>~2\text{ and }2 <spanclass=′latex−bold′>(B)</span> 2 and 0<span class='latex-bold'>(B)</span>~2\text{ and }0 <spanclass=′latex−bold′>(C)</span> 2 and infinitely many<span class='latex-bold'>(C)</span>~2\text{ and infinitely many} <spanclass=′latex−bold′>(D)</span> Infinitely many and 0<span class='latex-bold'>(D)</span>~\text{Infinitely many and }0
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