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Rational Round

Age 16 to 18
Challenge Level

Show that for every integer $k$ the point $(x, y)$, where

$$x = {2k\over k^2 + 1}, \ y = {k^2 - 1\over k^2 + 1},$$

lies on the unit circle, $x^2 + y^2 =1$. That is, there are infinitely many rational points on this circle.

Show that there are no rational points on the circle $x^2 + y^2 =3$.