15

Part of 2021 AIME Problems

Problems(2)

Source: 2021 AIME 1/15

S

be the set of positive integers

k

such that the two parabolas

y=x^2-k~~\text{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

AMCAIME2021 AIME I

Rent a g(n)f(n)

Source: 2021 AIME II #15

f(n)

g(n)

be functions satisfying

f(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 1 + f(n+1) & \text{ otherwise} \end{cases}

g(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 2 + g(n+2) & \text{ otherwise} \end{cases}

for positive integers

n

. Find the least positive integer

n

\tfrac{f(n)}{g(n)} = \tfrac{4}{7}