Problem:
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=knamb centimeters per hour, where k,a,b are integer constants. In a population where all organisms have 5 toes, logv=4+2logm; and in a population where all organisms have 25 eyes, logv=4+4logn (base 10). What is k+a+b?
Answer Choices:
A. 20
B. 21
C. 22
D. 23
E. 24
Solution:
Note that logv=logk+logna+logmb=logv=logk+alogn+blogm.
Plugging into the first equation with n=5 and the second equation with m=25:
logk+alog5+blogm=4+2logmlogk+alogn+blog25=4+4logn
As the equations must be true no matter how n and m vary, the logn and logm terms must cancel out in each equation respectively. This gives a=4 and b=2.
Substituting into the first equation gives:
logk+4log5+2logm=4+2logm→logk=4−4log5=4(log10−log5)=4log2=log16
So k+a+b = 16+4+2 = (C) 22.
The problems on this page are the property of the MAA's American Mathematics Competitions