Question 15
Let $S$ be the set of positive integers $k$ such that the two parabolas $$
y = x^2 - k
$$ and $$
x = 2(y - 20)^2 - k
$$ intersect in four distinct points, and these four points lie on a circle with radius at most $21.$ Find the sum of the least element of $S$ and the greatest element of $S.$
y = x^2 - k
$$ and $$
x = 2(y - 20)^2 - k
$$ intersect in four distinct points, and these four points lie on a circle with radius at most $21.$ Find the sum of the least element of $S$ and the greatest element of $S.$