2020 AMC 12A Problem 13

Question

There are integers

a,

b,

and

c,

each greater than

1,

such that

\sqrt[a]{N \sqrt[b]{N \sqrt[c]{N}}} = \sqrt[36]{N^{25}}

for all

N \gt 1.

What is

b?

2

3

4

5

6

Write the left side as

N

to the power

\dfrac{1}{a} + \dfrac{1}{ab} + \dfrac{1}{abc}

Set that exponent equal to

\dfrac{25}{36}

and peel off one nested variable at a time starting with

a = 2

The left side equals

N

raised to the exponent

\dfrac{1}{a} + \dfrac{1}{ab} + \dfrac{1}{abc},

which must equal

\dfrac{25}{36}.

Because

b,c\ge2,

this exponent is at most

\dfrac{7}{4a}.

If

a\ge3,

it is at most

\dfrac{7}{12}\lt\dfrac{25}{36},

so

a=2.

The equation becomes

\dfrac{1}{2b}+\dfrac{1}{2bc}=\dfrac{7}{36}.

Clearing denominators gives

c(7b-18)=18.

Set

d=7b-18,

so

d

is a positive divisor of

18.

Among

1,2,3,6,9,18,

only

d=3

makes

b=\dfrac{d+18}{7}

an integer. Hence

b=3

and

c=6.

Thus, B is the correct answer.

Problem 13 in other years