[15 points] Two distinct real numbers $r$ and $s$ are said to form a good pair $(r, s)$ if $$r^3 + s^2 = s^3 + r^2$$ (i) Find a good pair $(a, \ell)$ with the largest possible value of $\ell$. Find a good pair $(s, b)$ with the smallest possible value $s$. For every good pair $(c, d)$ other than the two you found, show that there is a third real number $e$ such that $(d, e)$ and $(c, e)$ are also good pairs. (ii) Show that there are infinitely many good pairs of rational numbers. Hints (use these or your own method): The function $f(x) = x^3 - x^2$ may be useful. If $(r, s)$ is a good pair, can you express $s$ in terms of $r$? You may use that there are infinitely many right triangles with integer sides such that no two of these triangles are similar to each other.
[15 points] Two distinct real numbers $r$ and $s$ are said to form a good pair $(r, s)$ if
$$r^3 + s^2 = s^3 + r^2$$
(i) Find a good pair $(a, \ell)$ with the largest possible value of $\ell$. Find a good pair $(s, b)$ with the smallest possible value $s$. For every good pair $(c, d)$ other than the two you found, show that there is a third real number $e$ such that $(d, e)$ and $(c, e)$ are also good pairs.\\
(ii) Show that there are infinitely many good pairs of rational numbers.
Hints (use these or your own method): The function $f(x) = x^3 - x^2$ may be useful. If $(r, s)$ is a good pair, can you express $s$ in terms of $r$? You may use that there are infinitely many right triangles with integer sides such that no two of these triangles are similar to each other.