Problem

Source: Romania National Olympiad 2023

Tags: algebra



Prove that: a) There are infinitely many pairs $(x,y)$ of real numbers from the interval $[0,\sqrt{3}]$ which satisfy the equation $x\sqrt{3-y^2}+y\sqrt{3-x^2}=3$. b) There do not exist any pairs $(x,y)$ of rational numbers from the interval $[0,\sqrt{3}]$ that satisfy the equation $x\sqrt{3-y^2}+y\sqrt{3-x^2}=3$.