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2021 AIME Problems
15
Rent a g(n)f(n)
Rent a g(n)f(n)
Source: 2021 AIME II #15
March 19, 2021
Problem Statement
Let
f
(
n
)
f(n)
f
(
n
)
and
g
(
n
)
g(n)
g
(
n
)
be functions satisfying
f
(
n
)
=
{
n
if
n
is an integer
1
+
f
(
n
+
1
)
otherwise
f(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 1 + f(n+1) & \text{ otherwise} \end{cases}
f
(
n
)
=
{
n
1
+
f
(
n
+
1
)
if
n
is an integer
otherwise
and
g
(
n
)
=
{
n
if
n
is an integer
2
+
g
(
n
+
2
)
otherwise
g(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 2 + g(n+2) & \text{ otherwise} \end{cases}
g
(
n
)
=
{
n
2
+
g
(
n
+
2
)
if
n
is an integer
otherwise
for positive integers
n
n
n
. Find the least positive integer
n
n
n
such that
f
(
n
)
g
(
n
)
=
4
7
\tfrac{f(n)}{g(n)} = \tfrac{4}{7}
g
(
n
)
f
(
n
)
=
7
4
.
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