2025 AMC 12A Problem 7

Question

In a certain alien world, the maximum running speed

v

of an organism is dependent on its number of toes

n

and number of eyes

m.

The relationship can be expressed as

v = k n^a m^b

centimeters per hour, where

k,

a,

and

b

are integer constants. In a population where all organisms have

5

toes,

\log v = 4 + 2\log m;

and in a population where all organisms have

25

eyes,

\log v = 4 + 4\log n,

where the logarithms are base

10.

What is

k + a + b?

20

21

22

23

24

Take logs:

\log v = \log k + a\log n + b\log m

Setting

n = 5

gives

b = 2

and

\log k + a\log 5 = 4

; setting

m = 25

gives

a = 4

Taking logarithms,

\log v = \log k + a\log n + b\log m.

With

n = 5,

this reads

\log v = (\log k + a\log 5)

+ b\log m,

matching

4 + 2\log m,

so

b = 2

and

\log k + a\log 5 = 4.

With

m = 25,

it reads

\log v = (\log k + b\log 25)

+ a\log n,

matching

4 + 4\log n,

so

a = 4

and

\log k + 2\log 25 = 4.

Then

\log k = 4 - \log 625

= \log\dfrac{10000}{625}

= \log 16,

so

k = 16.

Hence

k + a + b = 16 + 4 + 2 = 22.

Thus, the correct answer is C.

Problem 7 in other years